de Groot, Arnold

Edixhoven, Bas

Top, Jakob

English

A list is given of elliptic curves over Q having additive reduction at exactly one prime. It is also proved that for primes congruent to 5 modulo 12, no such curves having potentially good reduction exist. This enables one to find in a number of cases a complete list of all elliptic curves with bad reduction at only one prime

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   University of GroningenElliptic curves over the rationals with bad reduction at only one primeEdixhoven, Bas; de Groot, Arnold; Top, JakobPublished in:Mathematics of ComputationIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1990Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Edixhoven, B., de Groot, A., & Top, J. (1990). Elliptic curves over the rationals with bad reduction at onlyone prime. 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For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 04-06-2022MATHEMATICS OF COMPUTATION VOLUME 54, NUMBER 189 JANUARY 1990, PAGES 413-419 ELLIPTIC CURVES OVER THE RATIONALS WITH BAD REDUCTION AT ONLY ONE PRIME BAS EDIXHOVEN, ARNOLD DE GROOT, AND JAAP TOP ABSTRACT. A list is given of elliptic curves over Q having additive reduction at exactly one prime. It is also proved that for primes congruent to 5 modulo 12, no such curves having potentially good reduction exist. This enables one to find in a number of cases a complete list of all elliptic curves with bad reduction at only one prime. 1. INTRODUCTION This paper grew out of an attempt to find some examples of elliptic curves over Q with conductor p 2, for various small primes p. The conductor N of an elliptic curve over Q is an invariant which contains information about the reductions of the curve modulo primes. If the curve has good reduction modulo p, then p does not divide N; if the curve has multiplicative reduction at p, then p divides N exactly once. In case p > 5 and the curve has additive reduction at p, then p divides N exactly twice. Our main result is that in certain cases the only curves with conductor p2 are twists of elliptic curves of conductor p. In combination with work of Setzer [9] and Brumer and Kramer [2], this allows one to write down for a number of primes p a complete list of the elliptic curves over Q with good reduction outside p. In other cases, an extensive computer search provided us with many examples for primes p < 5000. These are given in the second half of this paper. There is no guarantee that this list is complete. 2. DIOPHANTINE RESULTS Suppose p is a prime number. Our problem is to find all elliptic curves over Q having additive reduction at p and good reduction at all other primes. Since all elliptic curves with good reduction away from 2 and 3 are known (see e.g. [ 1, pp. 123-134]), one may assume p > 5. Given an elliptic curve as desired, one can twist it over the quadratic extension of Q with discriminant p (compare [10, Chapter X, ??2 and 5]). The resulting curve has conductor a power of p as well. Moreover, one has: Lemma 1. Let p > 5 be a prime number and K the quadratic extension of Q unramified outside p. Up to K-twists, all elliptic curves over Q with good Received September 12, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 IG05; Secondary 1 1D25. The third author was supported by N.W.O. i 1990 American Mathematical Society 0025-5718/90 $1.00 + $.25 perpage 413 414 BAS EDIXHOVEN, ARNOLD DE GROOT, AND JAAP TOP reduction away from p are 1. the ones with multiplicative reduction at p; 2 3 ? 4. 2. those given by a Weierstrass model with discriminant +p , +p , or +p Proof. Given p and K as in the lemma and any elliptic curve E over Q, let A 1 and A2 be the discriminants of Weierstrass models of E and the K-twist of E, respectively. Then the difference of the valuations at p of l and A2 is congruent to 6 modulo 12. Since any elliptic curve E over Q has a global Weierstrass minimal model ([10, p. 226]) and no such curve has everywhere good reduction (e.g. [10, Exercise 8.15]), the lemma follows using the table on p. 46 of [1I]. o: Our main interest is in curves which do not come from the ones with mul- tiplicative reduction at p. Curves of conductor p are easily constructed by a method of Mestre [6]. In view of the lemma, this means that we can restrict ourselves to the following case. Assume one has a Weierstrass model with discriminant A, for some A E {+p 2, +p3, +p4 . The c4 and c6 invariants of this model satisfy the relation C4 - C6 = 1728A ([1, p. 36]). Furthermore, since we are only interested in curves with additive reduction, we assume that c6 0 modp . Then the relation implies that also c4 is divisible by p. Hence these invariants yield an integral solution of the equation 3 2 -2 pX _ y = 1728p A. Analyzing this, one obtains: Proposition 1. If p is a prime number congruent to 5 modulo 12, then the only integral solutions to pX3 _ e {+1728, +1728p, +1728p2} are the ones with Y =O. Proof. If pm - n = +?1728 for some integers m and n, then it follows that one of +1728 and -1728 must be a square modulo p. If p -5mod 12, neither is, hence the equation has no integral solution in this case. 3 _ 2 _2 A similar argument works for pX y _- + 1 728p : given a solution X = m and Y = n, one checks that both m and n are divisible by p. The integers m/p and n/p satisfy p2 (m/p)3 - (n/p)2 = + 1 728. Again one of + 1728 must be a square modulo p, which is not the case. If integers m, n satisfy pm - n = +?1728p, then it follows that ( ? ) 2 2 3 Hence we obtain a solution (x, y) EQxQ of dy2 = X3 + 1, for d = 3p. By an elementary descent argument, Nagell [7, pp. 48-53]) showed already in 1925 that the only rational solution to this is the one with y = 0. This proves the proposition. o ELLIPTIC CURVES OVER THE RATIONALS 415 Remark. In fact, more is true: the descent argument referred to is actually performed for dy2 = X3 + 1, where d is any integer not divisible by primes congruent to +1 or 7 modulo 12. (See also Nagell's beautiful review paper on the arithmetic of curves [8, p. 15].) Using the method of descent via two- isogeny, one can easily check this result of Nagell, since one does not come across nontrivial elements in a Shafarevich group (see e.g. [10, pp. 302-304]). The proposition immediately implies, using the discussion at the beginning of this section: Theorem 1. If p is a prime number congruent to 5 modulo 12, then every elliptic curve over Q with conductor p2 is a twist of one with conductor p. Corollary 1. For the following primes p < 1000 no elliptic curve over Q with good reduction at all places except p exists: 5, 29, 41, 137, 173, 257, 281, 293, 317, 401, 449, 461, 509, 521, 569, 617, 641, 761, 809, 821, 857, 881, 929, 953, 977. Proof. This follows from pp. 732-733 of [2] and our theorem. E Corollary 2. Suppose p is a prime number of the form u + 64 such that 3 t u. If both the class numbers of Q( -p) and Q(,J/5) are not divisible by 3, then there exists exactly one isogeny class of elliptic curves over Q with conductor p2 This is, e.g., the case for the following five primes: 113, 353, 3089, 4289, 9473. Proof. Combine our theorem with work of Setzer [9, p. 367]. o 3. NUMERICAL RESULTS As explained in the previous section, we are interested in finding Weierstrass 3 2 N models over Q whose c4 and c6 invariants are related by c4 - C6 = + 1 728p for 2 < N < 4. Given integers c4, C6 related like this, there is only one elliptic curve over Q with these invariants, namely the one given by y2 = x - 27c4x - 54c6. This model has discriminant +2 23 2pN , hence it defines an elliptic curve with conductor p2 if and only if it is not minimal at both 2 and 3. Minimality at 3 is easily checked: Lemma 2. Suppose c4 and c6 are integers satisfying c4 - c 2 = 1728d with 3 { d. Take a such that a-2c6 mod 3 and w = (a3 - 2c6)/3-ac4. The curve given by y2 = x3 - 27c4x - 54c6 is not minimal at 3 if and only if a - c4 is divisible by 3 and w is divisible by 9. This follows, e.g., from Tate's algorithm (see [1, pp. 47-52]). Minimality at 2 is also checked using Tate's algorithm, but requires a little bit more work. The strategy is now as follows. Given p, search for integers c6 for which c2 + 1728pN is a cube (2 < N < 4). (By solving this modulo powers of p, 2 one knows in advance the possibilities modulo p of a solution c6, and even 4 modulo p in case N = 4.) Having such a c6, compute the corresponding c4 and check whether the equation y2 = x3 - 27c4x + 54c6 is minimal at 2 and 3. 416 BAS EDIXHOVEN, ARNOLD DE GROOT, AND JAAP TOP If not, compute a global Weierstrass minimal model corresponding to this. This was done for all (relevant) primes p such that 7 < p < 5000 and all values Ic61 < 250000000. The program found 65 elliptic curves with conductor p2, given in the following list: A al a2 a3 a4 a6 C4 C6 73 1 -1 0 -37 -78 1785 75411 -73 1 -1 0 -2 -1 105 1323 - 112 1 1 1 -30 -76 1441 54703 -11 3 0 - 1 1 -7 10 352 -6776 -114 1 1 0 -2 -7 121 5203 -i193 0 0 1 -38 90 1824 -77976 -372 1 -1 1 2 -2 -111 999 -372 1 -1 1 - 1663 -25680 79809 22546431 373 0 1 1 -12 -17 592 10952 373 0 1 1 -2602 50229 124912 -44147512 -433 0 0 1 -860 9707 41280 -8387064 434 1 0 1 -39 -27 1849 20339 -472 1 1 1 1 -2 -47 1927 -673 0 0 1 -7370 243528 353760 -210408408 793 1 -1 0 -64 -179 3081 168507 -1573 1 1 0 -42 125 2041 -123245 -1633 0 0 1 -2174420 1234136692 104372160 -1066294102104 -1792 1 -1 0 -11 -14 537 14499 -1912 1 0 1 -92 -345 4393 291275 2772 1 0 0 -75 -256 3601 215783 3072 1 -1 0 -19 -24 921 24867 -3592 1 0 1 7 -15 -359 13283 3972 1 -1 1 - 174 924 8337 -761049 -3973 1 0 0 91 190 -4367 -157609 -4314 1 0 0 -3870 92773 185761 -80434513 4334 1 0 0 -3906 -93853 187489 80807759 -4432 1 0 1 -102 -403 4873 340667 -5032 1 0 1 10 21 -503 -17605 -6732 1 0 0 -14 37 673 -32977 7392 1 -1 0 -46 127 2217 -99765 -8632 1 -1 0 -701 7322 33657 -6174765 8773 0 -1 1 -2046 -34926 98224 30765160 -8872 1 1 0 19 -22 -887 2572 -1069' 1 1 1 -824 -9602 39553 7999327 -11032 1 0 0 23 -30 -1103 27575 -12132 1 0 1 -329 2265 15769 -1980829 -12592 1 -1 1 -79 -256 3777 237951 -12972 1 1 1 -27 -94 1297 71335 -13672 1 1 0 29 42 -1367 -25973 -14392 1 1 1 30 -16 -1439 24463 -15592 1 0 1 32 -21 -1559 20267 -16072 1 0 1 33 23 -1607 -17677 17532 1 1 0 -36 -5 1753 -8765 -17532 1 0 1 -37 -123 1753 103427 17772 1 1 1 -37 -30 1777 12439 -18793 1 -1 1 -9043 333258 434049 -285981921 ELLIPTIC CURVES OVER THE RATIONALS 417 A al a2 a3 a4 a6 C4 C6 -19072 1 1 1 -257882 50298256 12378337 -43550530865 -19932 1 0 1 -42 137 1993 -121573 19992 1 1 0 -291 1792 13993 -1653173 20172 1 1 1 -42 22 2017 -34289 20892 1 1 0 -43 -64 2089 39691 22512 1 -1 1 -141 668 6753 -546993 26892 1 1 1 -56 -120 2689 83359 29532 1 0 1 -62 -125 2953 103355 -32032 1 -1 0 -200 -1051 9609 951291 -32292 1 1 0 -10561 413370 506953 -360953765 -33132 1 0 0 -69 -278 3313 235223 33312 1 -1 0 -208 -1093 9993 989307 -34692 1 0 1 -940 11007 45097 -9577909 41292 1 0 0 -86 227 4129 -202321 -41392 1 1 0 -29921 1979690 1436233 -1721223845 -42012 1 1 0 -87 338 4201 -323477 44232 1 0 0 -645 6248 30961 -5444713 -45132 1 1 1 -94 -452 4513 356527 45973 1 1 1 -5842 -173646 280417 147926863 It should again be emphasized that there is no reason why this list should be complete. One way to find more examples would be to search for isogenies from the curves in the list to new ones. It turns out that there are no isogenies defined over Q from curves in our list to new ones, except in the cases where the curve has complex multiplication. In this case, the isogeny is from the curve in our list to its quadratic twist over Q( -p), so we do not list this isogenous curve. The search for isogenies can be done as follows. Any rational isogeny is built up from rational isogenies of prime degree. Such an isogeny of degree p corresponds to a rational point on the modular curve X0(p). If the genus of this modular curve is zero (which means p E {2, 3, 5, 7, 13}), a good model of it can be found on pp. 238-240 of [6]. In fact we only have to look at integral points on these models, since the curves in our list and those isogenous to them have potentially good reduction, hence integral j-invariants (compare [10, Chapter VII, Proposition 5.5 and the proof of Corollary 7.2]). In this way we find three isogenies: 1. The two curves with conductor 72 in the list are 2-isogenous; this cor- responds to the point (-1, -2 2) on the model xy = 212 of X0(2). 2. There is a 7-isogeny between the curves with discriminant -37 2; it corresponds to (-1, -49) on the model xy = 49 of XO(7). 3. Our curves with discriminant 373 are 5-isogenous. This corresponds to the point (1, 53) on the model of X0(5) given by xy = 53 In case the genus of X0(p) exceeds zero, all rational points are known by work of Mazur [5, pp. 129-130]. It turns out that such points only exist for p c {11, 17, 19, 37, 43, 67, 163}. We list the results; they do not give any new examples. 418 BAS EDIXHOVEN, ARNOLD DE GROOT, AND JAAP TOP 1. For p = 11 , the isogenies can be found on p. 97 of [1]. The new curves this provides are twists of curves in our table. 2. The curves which admit a rational 17-isogeny are given on p. 80 of [1]. They have multiplicative reduction at 2. 3. The case p = 37 is also given on p. 80 of [1]. No twist of these curves has bad reduction at only one prime. 4. The remaining cases correspond to elliptic curves having complex mul- tiplication. As remarked above, they are isogenous to a twist. (Compare [10, p. 265].) Using the methods developed in his thesis [3], B. Edixhoven computed for prime numbers p < 3500 the number of so-called 'Weil curves' of conductor p2 which have potentially supersingular reduction at p. He found exactly the same number of such curves as can be found in our list. However, we do not know whether these curves in our list are actually Weil curves. 4. COMPLEX MULTIPLICATION CASES An excellent reference for the theory of elliptic curves with complex multi- plication is [4]. The following result may be classical (compare loc. cit., p. 35, Theorem 12.2.1): Fact 1. For any imaginary quadratic field K with class number one, there exists an elliptic curve over Q whose endomorphism ring equals the ring of integers of K, and moreover it has good reduction outside the discriminant of K. Of course this can be checked by merely writing down a model as desired for each of the eight well-known j-invariants involved. A much more general argument can be found in [4]. The table above contains all the complex multiplication curves (correspond- ing to imaginary quadratic fields with discriminant p > 5). For convenience they are listed here again (compare p. 84 of [4]). A a I a2 a3 a4 a6 C4 C6 73 1 -1 0 -37 -78 1785 75411 -73 1 -1 0 -2 -1 105 1323 - 1 13 0 -_1 1 -7 10 352 -6776 - 193 0 0 1 -38 90 1824 -77976 -433 0 0 1 -860 9707 41280 -8387064 -673 0 0 1 -7370 243528 353760 -210408408 -1633 0 0 1 -2174420 1234136692 104372160 -1066294102104 ACKNOWLEDGMENT It is a pleasure to thank F. Oort for his interest in this work and for encour- aging us to publish the results. BIBLIOGRAPHY 1. B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Lecture Notes in Math., vol. 476, Springer-Verlag, 1975. 2. A. Brumer and K. Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), 715-743. ELLIPTIC CURVES OVER THE RATIONALS 419 3. S. J. Edixhoven, Stable models of modular curves and applications, Ph.D. Thesis, Math. Inst., Univ. of Utrecht, 1989. 4. B. H. Gross, Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Math., vol. 776, Springer-Verlag, 1980. 5. B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162. 6. J.-F. Mestre, La methode des graphes. Exemples et applications, Proc. Internat. Conf. on Class Numbers and Fundamental Units of Algebraic Number Fields (Nagoya Univ., Nagoya, 1986), Katata, Japan, 1986, pp. 217-242. 7. T. Nagel, Uber die rationale Punkte auf einigen kubischen Kurven, T6hoku Math. J. 24 (1925), 48-53. 8. M. T. Nagell, L'analyse indeterminee de degree superieur, Memorial des Sciences Mathema- tiques 39 (1929). 9. B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. 10 (1975), 367-378. 10. J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer- Verlag, 1986. MATHEMATICAL INSTITUTE, UNIVERSITY OF UTRECHT, BUDAPESTLAAN 6, 3508 TA UTRECHT, THE NETHERLANDS. E-mail: bas~math.berkeley.edu topj~qucdn 

Elliptic curves over the rationals with bad reduction at only one prime

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Elliptic curves over the rationals with bad reduction at only one prime

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