ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 21 Oct 2024 11:04:59 +0200How to find a CM elliptic curve with small integer coefficients?https://ask.sagemath.org/question/79122/how-to-find-a-cm-elliptic-curve-with-small-integer-coefficients/I want to find a CM elliptic curve with relatively small integer coefficients and complex multiplication discriminant equal to -163.
The coefficient of the elliptic curve found by 'EllipticCurve_from_j' is not small enough.
> cm_j_invariants(QQ)
> cm_j_invariants_and_orders(QQ)
> E0 = EllipticCurve_from_j(-262537412640768000)
> E1 = E0.short_weierstrass_model();E1 # y^2 = x^3 - 34790720*x + 78984748304
> E2 =EllipticCurve([-8697680,9873093538]) # y^2 = x^3 - 8697680*x + 9873093538
> [E2.has_cm(),E2.cm_discriminant()] # [True, -163]
I want to search for the short Weierstrass elliptic curve by brute force search.
$y^2=x^3-ax+b\quad(0 < a < b < 10000000000)$ , where a and b are both integers.
Here's a wrong brute-force search approach using SageMath:
> for a in range(1, 10000000000):
> for b in range(a+1, 10000000000):
> E = EllipticCurve([0, 0, 0, -a, b])
> if E.cm_discriminant() == -163:
> print(f"Found curve: y^2 = x^3 - {a}*x + {b}")
Thanks in advance!Sun, 08 Sep 2024 17:38:31 +0200https://ask.sagemath.org/question/79122/how-to-find-a-cm-elliptic-curve-with-small-integer-coefficients/Comment by yx7 for <p>I want to find a CM elliptic curve with relatively small integer coefficients and complex multiplication discriminant equal to -163.
The coefficient of the elliptic curve found by 'EllipticCurve_from_j' is not small enough.</p>
<blockquote>
<pre><code>cm_j_invariants(QQ)
cm_j_invariants_and_orders(QQ)
E0 = EllipticCurve_from_j(-262537412640768000)
E1 = E0.short_weierstrass_model();E1 # y^2 = x^3 - 34790720*x + 78984748304
E2 =EllipticCurve([-8697680,9873093538]) # y^2 = x^3 - 8697680*x + 9873093538
[E2.has_cm(),E2.cm_discriminant()] # [True, -163]
</code></pre>
</blockquote>
<p>I want to search for the short Weierstrass elliptic curve by brute force search.
$y^2=x^3-ax+b\quad(0 < a < b < 10000000000)$ , where a and b are both integers.</p>
<p>Here's a wrong brute-force search approach using SageMath:</p>
<blockquote>
<pre><code>for a in range(1, 10000000000):
for b in range(a+1, 10000000000):
E = EllipticCurve([0, 0, 0, -a, b])
if E.cm_discriminant() == -163:
print(f"Found curve: y^2 = x^3 - {a}*x + {b}")
</code></pre>
</blockquote>
<p>Thanks in advance!</p>
https://ask.sagemath.org/question/79122/how-to-find-a-cm-elliptic-curve-with-small-integer-coefficients/?comment=79738#post-id-79738The curve `E2` you've found does satisfy your condition `0 < a < b < 10000000000`. How much smaller would you like the coefficients to be?Mon, 21 Oct 2024 11:04:59 +0200https://ask.sagemath.org/question/79122/how-to-find-a-cm-elliptic-curve-with-small-integer-coefficients/?comment=79738#post-id-79738