American Journal of Computational Mathematics, 2012, 2, 358-365
http://dx.doi.org/10.4236/ajcm.2012.24049 Published Online December 2012 (http://www.SciRP.org/journal/ajcm)
On Elliptic Curves with Everywhere Good Reduction
over Certain Number Fields
Shun’ichi Yokoyama
Faculty of Mathematics, Kyushu University, Fukuoka, Japan
Received September 1, 2012; revised October 21, 2012; accepted November 6, 2012
ABSTRACT
We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quad-
ratic fields

m

for . These results of computations give best-possible data including structures of Mor-
dell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain
cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic
field using our method.
200m
Keywords: Elliptic Curves over Number Fields; Mordell-Weil Group; Two-Descent
1. Introduction
Tables of elliptic curves over  have been of great
value in mathematical research. In particular, some
databases are very famous and useful in number theory,
and Cremona’s index (classification of elliptic curves
over ) becomes popular. Nowadays, modularity
theorem that explains correspondence between elliptic
curves and modular forms becomes one of the most im-
portant facts in number theory and arithmetic geometry.

Meanwhile, computing elliptic curves (rank of curves,
Mordell-Weil groups etc.) over general number fields is
still hard. There are only a few databases of such curves
and these databases use Cremona-Lingham’s general
algorithm over number fields. It seems ideal from the
viewpoint of computational approach, and we can also
observe the case of elliptic curves over cubic fields that
are not totally real. However, updating of this algorithm
with supplementary tables had been stopped since Sep-
tember 2005. In addition, though we apply this general
algorithm, we have to determine many Mordell-Weil
groups (=sets of rational points) and this task is the most
difficult in creating databases.
Therefore, we have to find a more efficient way (re-
ducing the number of Mordell-Weil groups that we have
to determine) to achieve this project and also easy-to-
read sorted tables of such curves, including information
(with references) which case is already known and which
case is still open.
Let m
K
be the real quadratic field


m where m
is a square-free positive integer with and
m
200m
K
O the ring of integers of m
K
. We already know the
following results concerning elliptic curves with every-
where good reduction over certain real quadratic fields
[1-16]:
Theorem 1.1.
1) There are no elliptic curves with everywhere good
reduction over m
K
if
2, 3, 5,10,11,13,15,171, 23, 30, 31, 34, 35,
39, 42, 47,53,55,57,, 66, 69,70, 73,
74,78,82,83,85,89,,95 and 97.
m,19, 2
58,61
93,94

2) The elliptic curves with everywhere good reduction
over m
K
are determined completely for
6,7,14,22,29,33,,41,65 and 77.m37,38

3) There are elliptic curves with everywhere good re-
duction over m
K
if 26nd 86.m,79 a

We can also consider the pure cubic field case. Let
m
L be the pure cubic field


3
re m is be-
free, positive integer with 20m m
m whecu
and 0
L
O ring
of integers of m
L. e first known result is given by
Bertolini-Canuto [17]:
the
Th
Theorem 1.2. Let be the field
L


 where

is the real cube root of 2 (i.e. 2). Then there are no
elliptic curves over with good reduction everywhere.
LL
L
Recently, N. Takeshi applied Bertolini-Canuto’s me-
thod and showed the following criterion in her master’s
thesis.
Theorem 1.3. ([18]) Let L be the cubic field (not only
C
opyright © 2012 SciRes. AJCM

S. YOKOYAMA 359
pure cubic) satisfying the following conditions:
a) 2 does not split on L,
b) The narrow class number of is coprime to 6.
L
Then there are no admissible curves over L (= elliptic
curves with everywhere good reduction over L which
have L-rational point of order 2).
In this paper, we extend Theorem 1.1 and apply our
method to determine the existence and nonexistence of
elliptic curves having good reduction everywhere over
certain pure cubic fields. The following two theorems are
our main result:
Theorem 1.4.
1) There are no elliptic curves with everywhere good
reduction over m
K
if m = 43, 46, 59, 62, 67, 71, 103,
107, 127, 137, 139, 151 and 163.
2) The elliptic curves with everywhere good reduction
over m
K
are determined completely for . 109m
3) There are admissible curves over m
K
if m = 118,
134, 161 and 166.
4) There are no admissible curves over m
K
if m =
131, 179 and 199.
5) There is an elliptic curve with everywhere good
reduction and not having
E
m
K
-rational point of order 2 if
m = 158 and 161.
Theorem 1.5.
1) There are no elliptic curves with everywhere good
reduction over if
m
L
3,5,6,10,12,17,18,29,116,137,173 and 197.m
2) If , there are no ad-
missible curves and elliptic curves with everywhere good
reduction over which have cubic discriminant.
23,44,45,75 and 87m
L
E
m
3) There is an elliptic curve with everywhere good
reduction and not having -rational point of order 2
over if .
m
L
m
We would like to remark that this result is an exten-
sion of the author’s previous result [19].
L46m
2. Strategy
In this section, we introduce the strategy to prove our
results. Our strategy for the proof is close to that of T.
Kagawa [7]. However, we use different kinds of com-
puter softwares and computational techniques.
Important processes of our result are the following: At
first, we divide all elliptic curves having everywhere
good reduction into two types. One is “admissible case”,
and the other is “nonadmissible case”. Next we consider
some criteria of S. Comalada to determine whether ad-
missible curves exist or not (Section 2.1). After that, we
observe (non)existence of all nonadmissible curves using
some criteria from algebraic number theory (Section 2.2).
Using this method, we can get the list of important in-
variants having constraint condition, and this condition
can be expressed using certain elliptic curves over m
K
or m. Finally, we directly compute Mordell-Weil
groups of specific elliptic curves (Section 2.3). Explicit
data are given from Section 2.4 to 2.6. We note that
easy-to-read sorted tables will be given in Chapter 3.
L
2.1. For the Case of Admissible Curves
First of all, S. Comalada [20] determines all admissible
curves defined over m
K
with . Comalada also
gives some criteria to find admissible curves over
100m
m
K
for an arbitrary . m
Definition 2.1. An elliptic curve defined over m
K
is
called g-admissible if it is admissible and has a global
minimal model.
Proposition 2.2. The following two conditions are
equivalent:
1) There exists a g-admissible elliptic curve over m
K
.
2) 1023m

or either of these sets of diophantine
equations has a solution:
a) 22
47xmy

 , , 7|m
b) 22 46xmy 5

, , 65|m
c) 22
2xmy

 , ,

2mod 8m
d) 22
8xmy

 and , is odd,
22
256rms r


od 81mm.
e) and ,
22
16384rms 22 8tmw r


od 4r3m ,


,1tr

, ,

128 modwst r


1mod 8m.
Thus we can find some admissible curves appearing
Theorem 1.4 using Comalada’s method.
2.2. For the Case of Nonadmissible Curves
Next we assume that a number field K is m
K
or .
We also assume that the class number of K is 1 and every
elliptic curve E with everywhere good reduction over K
has no K-rational point of order 2. For our convenience,
we say “nonadmissible” if E has everywhere good re-
duction over K with no K-rational point of order 2. First
we use the following result:
m
L
Proposition 2.3. ([21]) Let E be an elliptic curve over a
number field K. If the class number of K is prime to 6
then E has a global minimal model.
Let E be an elliptic curve with everywhere good re-
duction over a number field K. By Proposition 2.3, E has
a global minimal model
232
13 24
:Eyaxy ayxaxax a
6

 
with coefficients


1, 2, 3,4, 6
iK
aOi
E
.
The discriminant of (denoted by


E) is

32
46
1728
cc
E 

S. YOKOYAMA 360
46
,
K
cc O
as polynom
where are, as in [22] (Chapter III, p. 42),
written ials in the ’s with integer coeffi-
i
a
cients. Moreover, the following conditions are equivalent
(cf. [22], Chapter VII, Prop. 5.1):
1) E has everywhere good reduction over K,
2)

K
EO

.
In our case, all elements of
K
O are written in
rm n
the
fo

 where

is a fundamental unit of ( Klet us
fix

for each m Thus to determine the elliptic curves
with everywhere good reduction over K, we shall com-
pute the sets


).

,
nK
EO
xy




23
17280
n
KK
OO yxn
  
However, the set of coefficients
12.

5
12346
,,,,
K
aa aa aO

, which gives rise to

2
46
,
K
cc O

, does not necessarily exist. Therefore, we
check whether the curv
er K, has trivial conductor for
each
Actu hard to compute all
e
23
46
:2754
C
Ey xcxc,
which is isomorphic to E ov
 
46
,nK ccEO

.
ally, it is very


EO

hat some
results.
nK
because of the limitation of efficiency of equipments. To
reduceom the amount of cputation, we show t
values of n are irrelevant by using Kagawa’s
Before that, we can easily reduce for the cases of
612n because the map
 



23
6,, , nKnK
EOEOxyx y





ion. Hence, we obsis a bijecterve only for
whether
erywher
duct

nK
EO

06n.
In [7], Kagawa shows a criterion the dis-
criminant of an elliptic curve with eve good re-
verion o m
K
is a cube in m
K
:
Lemma 2.4. If the following five conditions hold, then
the discriminant of every elliptic curve with everywhere
good reductiover n om
K
is a cube in m
K
:
1) The class number of m
K
is prime to 6;
2) m K is unramified at 3;
3) The class number of


K3 is prime to
m 3;
he cla4) Tss number of


3
m
K

is odd;
P of 5) For some prime idealm
K
dividing 3, the con-
gruence

32
mod
X
P  ot have a does nsolution in
m
K
O.
Using wa shows the following:
Lemm 107,127,161,166 or 193
the criterion, Kaga
a 2.5. ([23]) If m

,
with everywhere good reduction over every elliptic curve
m
K
has a global minimal model whose discriminant is a
be in m cu
K
.
Therefore, we have


3n
E


 for some n

.
y applying the next lemma, we can further discard
some cas
B
es:
et K be a nund En
el
has no K-rational point of order 2, then
Lemma 2.6. ([7]) Lmber field a a
liptic curve defined over K. If E has good reduction
outside 2 and
 



2
K
EKE is a cyclic cubic extension un-
ramified outside 2. In particular, the ray class number of



K
E modulo 2PP M is a multiple of 3.
Note that




n
KEK


is either K,


1K

or


K


. Thus we compute the ray class
number of



K
E modulo
computan Table s
(Same typee o
26]). The um
M. The following
tions i 1 and 2 are carried out by using
Pari/GP [24] results werbtained in [25] by
using KASH [bold-faced nbers in this table
are the ones divisible by 3.
Remark 2.7. Using Lemma 2.6 with some arguments,
we conclude that if the class number of K is 2 and the
ray class numbers of K,

1K and


K


are
of Table 1. Ray class number


KE m. odulo
Ray class number
M
m


1K

m
K



m
K
m



43 3 1 10
46 4 1 3
103 5 20
107
15 6
12
15
59 9 6 1
62 8 3 1
67 3 14 1
71 7 3 4
1
1 9 6
109
127 5
3 1 1
3 16
131 1
137 4 1 1
139 9 14 1
151 7 1
161 8 3 1
163 3 22 3
179
193 2
1
1
6
1
199 9 20 1

S. YOKOYAMA 361

 m
LE Table 2. Ray class number of

modulo
Ray class number
M.
m



1
m
L

m
L



m
L


M1 1 1 1
M2 3 1 1
M3 1 3 1
M1 3
M5 3 3 1
4 1
M6 3 1 3
M7 6 1 1
M8 21 1 1
k
M
’s appearing in the above are aollows:

1k8 s f


12,6,10,12,17,18,216,137,173,197M
3,5, 9,1,


2244, 46,53, 71, 829,145,167,179,M3,33, ,9


34 ,M
5,87


475 ,M


541,55 ,M


659,69,188,
M


7107 ,M


8177 .M
all prime to 3 everywhere
good reduction over K is admissible (See [25], Cor. 2.3).
In this way, we compute them for
then each elliptic curve with
m
K
with m = 118,
134, 158, 16 and wenclude thit cannot
termined for these 4 cases whether tere is an elliptic
curve with everywhere good reduer
6 coatbe de- the
h
ction ovm
K
which
ave noh m
K
-rational point of ordeMewhile, we
w
r 2. an
can show the (non)existence of an admissible curve (see
the next section).
For example, the case of 46me can conclude
that

,


mm
KEK

  thus the discriminant
has the form


21
0
k
Ek 


. Hence should
determine three sets of integral points

1m K
EO
,

E

we
omputing Mordell-We
pute the dell-
3m K
O and

5m K
EO
.
2.3. Cil Basis and Integral
Points
To compute

nK
EO
, we first comMor
Weil group




23
,172806.
n
xy K Kyxn   
is decom
n
EK

It posed into a direct-sum of (tor-
it rsion pa deter-
mined by observing reduction at good primes and de-
composition of division polynomials. On the other hand,
the free part can be computed by applying two-deent
and infinite descent (the process of decompression from

ntors
EK

rt can be
sion part) and r
free

 (free part, which is not
canonical, w

n
EK

h 0r). The to
sc




2
nn
K EK to E



n
EK). We usedmon’s
two-descent program . [27]) on Pari-GP [24]. To com-
pute some related data efficiently, we executed the Pari-
GP program on Sage [28] as a built-in software. We also
use Magma [29] for verification.
The procedure of explicit computation of

(cf
Denis Si


EK is
the following:
1) Determine 1,,ppose images in
r wh




tors
EK EK generate a subgroup of finite index of




tors
EK EK. Usually, these are obtained by per-
forming an m-descent for some 2m, especially we
often choose 2m

.
2) Compute an upper bound on the index:




1
p:,,
r
tors
K EKpE



. 
3) A sieving
used to deduce a M
We certainly w
procedure (See [30], Section 4) is then
ordell-Weil basis for

EK.
ish to have an upper bound for




1
:,,
r
r
EK EKpp
to s




 as as possible. In
pa
small
ordell-Weil
basis of
rticular, 1,,
r
pp will certainly be a M


EK if the value is equal to 1.
To coof integral points in mpute the subset

nK
EO



nK
, we use the method of elliptic logarithm to
compute the linear form:
E

r

 
1,, , K r
mmn
1ii n
i
LmpnTEO


where i
p’are generators of the free part and
the torsio. Moreover, the maximum o
s and
partf the absolute
va the line
T
n
lues of the coefficients ofar form


1
max, ,,
r
M
mmn
can be bounded using the LLL-algorithm (by Lenstra-
Lenstra-Lovasz, cf. [31]).
ly p
Fible curves:
Proposition 2.8.
1) There are no g-admissible curves over
Final, we comute that the elliptic curve
23
46
:2754
C
Ey xcxc
has trivial conductor.
2.4. Computation I: Admissible Curves for Real
Quadratic Case
rst we prove the (non)existence of g-admissi
m
K
if m =
1, 137, 139, 151, 15, 179,
and 199.
103, 107, 109, 127, 138, 163
2) There are g-admissible curves over m
K
if m = 118
161 and 166.
,
134,
Proof. For all m’s appearing in 1), the third equivalent
conditions a)-e) of Proposition 2.2 does not be satisfied.

S. YOKOYAMA 362
For 118, 134m and 166, we can find a solution of
2)
n get the
-c) of Prop. 2.2. The equation has the form
22
2xmy
and we cafollowing solutions:
(Case 118m) 22 554118512

, 
(Case 134m) 22 382134332

,
Case 166m) 22 4124 (216632012 .
For 161m, we can find a solution of 2)-a) of Prop.
2.2. The equation has the form
and we can get the following solution
Rem9. In fa20] proved that the numer of
g-sible e
22
47xmy
2
2 2034161 87.
ark 2.ct, [b
admislliptic curves over m
K
(up to isomor-
phism) for 118,134, 166m is 2
lude that the num-
ber of admissible elliptic curves over
if
22
2xmy
is solvable and 6m. Thus we conc
m
K
ote
for m = 118,
134, 166 is gror equal to 2. that it is not
tru
eater than N
e in general that all admissible curves defined over
m
K
are g-admissible. However, asse the class num-
ber of m
um
K
is odd, it is true except som cases.
2.5. Computation II: Nonadmissible Curves for
e
Real Quadratic Case
Proposition 2.10. If m = 43, 46, 59, 62, 67, 71, 107, 127,
139, 151 and 163, there are no elliptic curves with every-
where good reduction over m
K
.
Proof. We compute Mordell-Weil bases and the sets of
integraints for each of the 11 cases usingl po method
f
o
avoi ailable from
curve w
appearing in Section 2.3. In this paper, we omit data o
bases of

nm
EK
 and the sets of integral points t
d being intricate. A complete data are av
the author’s website:
http://www2.math.kyushu- u.ac.jp/~s-yokoyama/ECtab le.
html (*).
As a result, there are no pairs 

46
,m
nK
ccE O

 for
which the elliptic curve C E has trivial conductor.
For the case m = 109, we can find

46
,cc (that gives
the elliptic ith everywhere good reduction appear-
ing in Theorem 1.4) from


109
4K
EO
.
Proposition 2.11. The elliptic curves with everywhere
good reduction over
m
K
are determined completely for
m109.
Proof. First, we compu Mordell-Weil bases: te
1)

0 1092EK
 ; basis is


 
10 12,0TT .
2)

2
EK

 ; basis is

,pp where

9 109
2109 109 109AB




307444 12529452 125109 ,
109 5688 5 A
p 25544 2109,




7109
109 3026 9290 9109, B
p
.
277340 2726564 2
3)


2
4 109
EK


 ; basis is where

109 109
,
CD
pp


109 6 109,1296 109p55965386204 1231 C ,




109 916346 818777081109,
1613792380 729154573276 729109
D
p
.
The sets of integral points are
1)



109
0 109
,
K
EO OT
,
2)



109
2
109 109109109109109
,,2,22
K
ABABAB
EO
Oppp pp p  
,




109
4109 109 1099
,,2,3
KCCC
EOOpppp
 . 3) 10 D
From in
109
2C p


109
4K
EO
, we can construct the elli-
ptod reduction over ic curve having everywhere go109
K
as follows:

23
1 1093 109
22
27429 1093259315109.
yxyxx
x


 
2
ording to [16], there are no elliptic curves having
goeduction everywhere and no
Acc
od r109
K
-rational poi
er 2 (-admissib
nt
of ord= nonle) except the above up to
isogenies.
For the case 158m

, the class number of m
K
is 2
gypply. However, we can one
elliptic curvehere good reduct
so our strate cannot a
with everyw
find
ion over m
K
with computing


158
3K
EO .

Proposition 2.12. There is an elliptic curve having
everywhere good reduction over
E
158
K
. E is n by give

2
32
158
158 158
yxyy
xx ABxCD


where
361817559192191668 851A

,
287846594758031 45415B

,
3691288333C19186381273841 7681108

,
29366313214636 764917584806 2813D

.
For the case 161m

, we can find

46
,cc (that
gives the elliptic curve with evererood reductionywhe g

S. YOKOYAMA 363
appearing in Theorem 1.4) from
Proposition 2.13. There is an elliptic curve ving
everywhere good reduction over

161
3K
EO
.
E ha
161
K
. E is n by give



232
3680290 161
14848211702161 .
yxyyxxx 
 
2.n III: Curve
ere are no
el o
Proof. In this case, it is enough to determine
6. ComputatioNonadmissible s for
Pure Cubic Case
Proposition 2.14. If 23,44,7m th5 and 87,
t ord
liptic curves with everywhere good reductionver m L
which have no m L-rational poinf oer 2 (not ad-
missible) and cubic discriminant.
  


23 44 45 75 87
00333
,,,,
LLLLL
EO EO EO EO EO

.
(*). e no pairs
The result of computing Mordell-Weil bases and the
sets of integral points is available from the online data
As a result, there ar



46
,ccEO

m
nL
for which the elliptic curve C
E has trivial conductor.
Finally, to complete the proof of Theorem 1.5, we
show the existence of an elliptic curve having every-
wh ere good reduction over 46 L. We can find


46
,cc
from
Proposition 2.15. The elliptic curve E as follows is
ha

.
46
0L
EO
ving everywhere good reduction and not admissible
over 46
L:

323
234646 1
323
32 312 3
46 46
46 13
CCC
33
45 6
46 46
2
46 3 yx y y
3
x
xx

 
CC C

C4 = 23258423334479295709473275474986025640457
867,
C5 = 827892116462926667504946133778759990377913
857,
C6 = 326497412111533344905526205920140161442668
6175.
f. Wcan easily iminant of the
curve and the result is


where k C’s

16k are as follows:
1
C94219593757433390681493864706,
2108133470918663218473194 7617604,C
350840870355438304 371288085501 19,C
Prooe compute the discr

24
E


323
309 4648 464139  
where is a ndamental
fined over certain number fields. We note that
e,
PEX = Partly existence,
PNEX = Partly nonexistence,
ET = Undetermined,
is generator of each base field, given by
Magma’s setup “K<a>: = Number Field (f);” where f is
nomial of K.
4.
In he (non)existence of elliptic
cu reduction over certain
reases and partly determined
Existence result
fu
unit of 46
L.
3. Tables
We give Tables 3 and 4 showing the existence or
nonexistence of elliptic curves with everywhere good re-
duction de
DET = Determined,
NEX = Nonexistenc
UND
and a a
defining poly
We remark that precise version of the following tables
are available from the author’s website (*). These contain
data of fundamental units and references.
Conclusion
this paper we proved t
rves having everywhere good
al quadratic fields for 14 c
the (non)existence for 9 cases. We also proved such re-
sults over certain pure cubic fields for 12 cases and par-
tially proved for 6 cases.
Table 3. The case of real quadratic fields m K.
m
AdmissibleNon admissible Progress
11
mS

No No NEX
22
mS

Yes Yes DET
33
mS

Yes No DET
44
mS

No Yes DET
55
mS

Yes No PEX/UNDET
mS
66

No Yes PEX/UNDET
7
m7 S

Yes Yes PEXDE/UNT
88
mS

Yes - PEX/UNDET
99
mS

No - PNEX/UNDET
PN
10 10
mS - No EX/UNDET
where

12,3,5,,11,13,19, 21, 23,30,35,37,
39, 42, 43, 46, 47, 57,58, 59, 666, 67,
69,70,,73,74,2,83,85,89,9,97,
101,103,107, 113,127,129,137,139,141, 149,151,
163,16 173,17791,
S

197 ,
105,17,131, 34,
,53,551, 62,
7178,83, 94, 95
7, ,181,1


238,77S,


34, 22, 41, 65S6, 7,1,


429,33, 09,133,15S17 ,


5118S,


626, 7958S, ,1


786,161S, 


8134,166S,


951,87,91, 131, 17S, 9,199


10 193S.

S. YOKOYAMA 364
Table 4. The case of s
Existence result
pure cubic fieldm L.
m Admible Nonissible iss admProgress
11
mT No No NEX
22
mT No - PNEX/UNDET
46 No Yes PEX/UNDET
107 - - UNDET
where


12, 3, 5, 6,12,17,18, 29,116,137,1797T, 10, 3,1


223,33,41, 44,45,53,5559,69,71,752,87,99,145,167,177,179,188T. , ,8
It seems extremely difficult toxtendhese results us-
ing samof com-
putationese re
sults, we need to discoverrithms or new
mathatical t
5. Acknowledgements
The author likes to express his
gawaasanari Kida and Yuichiro Taguch
hime theoryf elliptic curves. Their advice during
eparation of his papers on elliptic curves was very
helpful.
The author would like to thank John Cremona an
enis Simon for making their programs available to him
an
pported by Grant-in-Aid for JSPS
fellows.
03-312. doi:10.1080/10586458.2007.10129002
e
om
t
e approach because of se limitations
al tools and equipments. To improve th-
efficient algo
emools.
gratitude to Takaaki Ka-
i whded , M
to th
o gui
o
the pr
d
D
d for initially guiding him through the intricacies of the
codes. He also would like to thank Iwao Kimura, Kazuo
Matsuno, Yu Shimasaki and Yukihiro Uchida who gave
him some useful advice and information.
This work was su
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