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Instantiating Elliptic Curve Isogenies using rational maps

asked 2021-10-13 21:34:29 +0100

RKnebel gravatar image

updated 2021-10-15 18:00:07 +0100

I have an elliptic curve $$E: y^2 = x^3+x$$ and the rational maps for a degree 14 isogeny $\phi =(\phi_x,\phi_y)$. I would like to get an Isogeny object using $\phi_x,\phi_y$ so I can use some of the functionality of the Isogeny class, but I can't seem to find anything in the sagemath documentation about this.

Currently sage allows you to construct either cyclic isogenies by specifying a point $P$ on $E$; or by the kernel polynomial provided the degree is either odd or the polynomial is divisible by $x^3+x$.

EDIT: I've looked the the code of the EllipticCurveIsogeny class and the attributes for the x and y rational maps are private attributes. I cannot figure out a way to create an Isogeny object with my desired rational maps.

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answered 2021-11-03 22:52:15 +0100

Luca gravatar image

This is not supported. Your best option is to pass the kernel polynomial, i.e., the denominator of $\phi_x$, as lone paramter to E.isogeny(). This will not give you exactly the isogeny with $\phi_x,\phi_y$ as rational maps, however you can compose that with an isomorphism to get the wanted isogeny.

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Asked: 2021-10-13 21:34:29 +0100

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Last updated: Nov 03 '21