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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
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pari:K = nfinit(Polrev(%s));
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magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R!%s);
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oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx(%s))
Generator \(a\), with minimal polynomial
\( x^{2} - x + 2 \); class number \(1\).
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sage:E = EllipticCurve([K([0,0]),K([1,-1]),K([0,0]),K([-1790,1493]),K([-12632,31268])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 12544.5-g have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 2 & 8 & 4 & 4 \\
8 & 1 & 4 & 4 & 8 & 2 \\
2 & 4 & 1 & 4 & 2 & 2 \\
8 & 4 & 4 & 1 & 8 & 2 \\
4 & 8 & 2 & 8 & 1 & 4 \\
4 & 2 & 2 & 2 & 4 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 12544.5-g over \(\Q(\sqrt{-7}) \)
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sage:E.isogeny_class().curves
Isogeny class 12544.5-g contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 12544.5-g1
| \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 1493 a - 1790\) , \( 31268 a - 12632\bigr] \)
|
| 12544.5-g2
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -278 a + 100\) , \( -1637 a + 2594\bigr] \)
|
| 12544.5-g3
| \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 93 a - 110\) , \( 524 a - 200\bigr] \)
|
| 12544.5-g4
| \( \bigl[0\) , \( -a\) , \( 0\) , \( -149 a - 185\) , \( -1884 a + 488\bigr] \)
|
| 12544.5-g5
| \( \bigl[0\) , \( a\) , \( 0\) , \( -37 a + 39\) , \( -20 a + 380\bigr] \)
|
| 12544.5-g6
| \( \bigl[0\) , \( -a\) , \( 0\) , \( -9 a + 95\) , \( -260 a - 72\bigr] \)
|
