Welcome to MathOverflow

Question

1

Let $E/\mathbb{Q}$ be an elliptic curve having a rational 3-torsion point. Then $E$ can be given an affine equation of the type $$y^{2} = x^{3} + (ax + b)^{2}$$ for $a, b, D \in \mathbb{Q}$. Has there been any work done about curves of this form having prime or almost prime conductors?

edited Jan 29 at 19:39

asked Jan 29 at 19:10

lk728
61●2

Actually that's the form of the general elliptic curve with a rational $3$-isogeny. For $3$-torsion, you can take $D=1$, or equivalently write the curve as $y^2 + a_1 x y + a_3 y = x^3$ with a $3$-torsion point at $(x,y)=(0,0)$. Do you want a $3$-torsion point or just the rational $3$-isogeny? – Noam D. Elkies Jan 29 at 19:20

3

I think it's known that the only $E / \mathbb Q$ of prime conductor with a $3$-torsion point, or even a rational $3$-isogeny, are the ones of conductor $19$ and $37$. What do you mean by "almost prime"? – Noam D. Elkies Jan 29 at 19:23

@Noam Elkies: I mean a 3-torsion point, so I've edited the question to reflect this. By "almost prime" I had in mind that the conductor is of the form $pq$ or $pqr$; forcing $E$ to be semistable and somehow restricting the number of factors might do that (though that is also something I'm not clear about). Do you have a source for the prime conductor? – lk728 Jan 29 at 19:43

5

Curves with prime-power conductor and $\ge 3$ torsion points were classified by Miyawaki: Osaka J. Math. 10 (1973) 309-323, ir.library.osaka-u.ac.jp/dspace/bitstream/11094/… . (Also, if you've got square-free conductor and an isogeny $E\to E'$ of prime degree $p>2$, then either $E$ or $E'$ has an actual $p$-torsion point, so his result gives you a complete classification for prime conductor, even if you only have a 3-isogeny rather than a 3-torsion point.) – Tim Dokchitser Jan 29 at 23:25

1

Miyawaki's work was generalized by Hadano in a paper in Manuscripta Math. in 1982. His Table I classifies (in some sense) curves with a $3$-torsion point and bad reduction at $2$ and a prime $p$. One can do something similar (and more precise) with $2$ replaced by $3$. In either case, the arguments extend easily to isogenies rather than torsion points. – Mike Bennett Jan 31 at 3:46

Elliptic curves with almost prime conductor

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.