Remarks on the conductor of an elliptic curve

Hadano, Toshihiro

doi:10.3792/pja/1195519724

Cited by 3 publications

(8 citation statements)

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“…are (x, y, h, n) = (7,3,5,4) and (x, y, n) = (2 h−2 − 1, 2 h−2 + 1, 2). Now we use some techniques in elementary number theory to find the integer solutions of some Diophantine equation.…”

Section: Diophantine Equationsmentioning

confidence: 99%

“…It was shown in [5] that all elliptic curves with conductor 2 m p n where p ≡ 3 or 5 mod 8, p = 3, that have a rational point of order 2, are effectively determined under the truth of the conjecture of Ankeny-Artin-Chowla.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.1 (Theorem 5.3.2, [10]). Let E be an elliptic curve over Q with prime conductor p. Then either |∆ E | = p or p 2 , or else p = 11 and ∆ E = 11 5 , or p = 17 and ∆ E = 17 4 , or p = 19 and ∆ E = 19 3 , or p = 37 and ∆ E = 37 3 . In particular, ∆ E | p 5 .…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, either x + 1 = ±2 h = ±2 2n−h−2 × 125 + 2, or x + 1 = ±2 h × 125 = 2 2n−h−2 + 2. Therefore, h = 1 or 2n − h − 1 = 1 which is a contradiction in both cases.✷ Let E/Q be an elliptic curve such that E(Q)[5] = {0}. Assume moreover that N E = p α q β where p = q are primes, and α, β > 0.…”

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confidence: 99%

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On elliptic curves whose conductor is a product of two prime powers

Sadek

2013

Math. Comp.

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We find all elliptic curves defined over Q that have a rational point of order N, N ≥ 4, and whose conductor is of the form p a q b , where p, q are two distinct primes, a, b are two positive integers. In particular, we prove that Szpiro's conjecture holds for these elliptic curves.problem under the condition that E has a rational torsion point. For example,in [6] the elliptic curves with conductor p m , p is prime, and 2-torsion points were listed.It was shown in [5] that all elliptic curves with conductor 2 m p n where p ≡ 3 or 5 mod 8, p = 3, that have a rational point of order 2, are effectively determined under the truth of the conjecture of Ankeny-Artin-Chowla.It is worth mentioning that the complete list of elliptic curves with a prime conductor has already been produced. The following theorem gives this list explicitly.Theorem 1.1 (Theorem 5.3.2, [10]). Let E be an elliptic curve over Q with prime conductor p. Then either |∆ E | = p or p 2 , or else p = 11 and ∆ E = 11 5 , or p = 17 and ∆ E = 17 4 , or p = 19 and ∆ E = 19 3 , or p = 37 and ∆ E = 37 3 . In particular, ∆ E | p 5 .The elliptic curves in Theorem 1.1 turn out to satisfy Szpiro's conjecture which is stated below for the convenience of the reader.Conjecture 1.2. If E is an elliptic curve over Q, thenOne of the popular strategies to find elliptic curves E/Q with a given conductor is to solve certain Diophantine equations obtained by equating the discriminant of E to the product of powers of the prime divisors of the conductor.Mazur gave a complete classification of the torsion subgroup E tors (Q) of E(Q), see (Theorem 7.5, §8, Chapter VIII, [11]). More precisely, E tors (Q) is isomorphic to one of the following fifteen groups: Z/nZ, 1 ≤ n ≤ 12, n = 11; or Z/2Z × Z/2nZ, 1 ≤ n ≤ 4.Given that P ∈ E(Q)[m], m = 2, 3, it is known that there exist b, c ∈ Q such that the following Weierstrass equation defines an elliptic curve E b,c isomorphic to E E b,c : y 2 + (1 − c)xy − by = x 3 − bx 2 with the image of P being (0, 0). The discriminant ∆(b, c) of E b,c is given as follows:By taking m to be an integer in {4, 5, 6, 7, 8, 9, 10, 12}, one finds an explicit relation between b, c, see for example §2 of [7].In this article, we generalize Theorem 1.1 to elliptic curves whose conductors have two distinct prime divisors only. More precisely, we generate the list of all elliptic curves

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Section: Diophantine Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On elliptic curves whose conductor is a product of two prime powers

Sadek

2013

Math. Comp.

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“…This was achieved for sets S of size at most 2 and assuming certain conditions on the elliptic curves. One may consult [6,7,8,14,16,17,19] in which there are lists of elliptic curves with at most two bad primes assuming the existence of a rational torsion point. The main idea is to solve certain Diophantine equations obtained by equating the minimal discriminant to a product of at most two prime powers.…”

Section: Introductionmentioning

confidence: 99%

Counting elliptic curves with bad reduction over a prescribed set of primes

Sadek

2017

Preprint

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Let p ≥ 5 be a prime and T a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over Q up to height X with Kodaira type T at p. This enables us find the proportion of elliptic curves over Q, when ordered by height, with Kodaira type T at a prime p ≥ 5 inside the set of all elliptic curves. This proportion is a rational function in p. For instance, we show that p 8 (p − 1) p 9 − 1 of all elliptic curves with bad reduction at p are of multiplicative reduction. Furthermore, we prove that the prime-to-6 part of the conductors of a majority (= ζ(10)/ζ(2) ≈ 0.6) of elliptic curves are squarefree, where ζ is the Riemann-zeta function.

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