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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 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
\( x^{2} + 2 \); class number \(1\).
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sage:E = EllipticCurve([K([0,0]),K([-1,1]),K([0,0]),K([-160,-32]),K([-680,-272])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 6912.3-j have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 8 & 4 & 2 \\
8 & 1 & 8 & 4 & 2 & 4 \\
4 & 8 & 1 & 8 & 4 & 2 \\
8 & 4 & 8 & 1 & 2 & 4 \\
4 & 2 & 4 & 2 & 1 & 2 \\
2 & 4 & 2 & 4 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 6912.3-j over \(\Q(\sqrt{-2}) \)
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sage:E.isogeny_class().curves
Isogeny class 6912.3-j contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 6912.3-j1
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -32 a - 160\) , \( -272 a - 680\bigr] \)
|
| 6912.3-j2
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -96 a + 96\) , \( 148 a + 844\bigr] \)
|
| 6912.3-j3
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 28 a - 40\) , \( -140 a + 52\bigr] \)
|
| 6912.3-j4
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 4 a + 56\) , \( -104 a + 16\bigr] \)
|
| 6912.3-j5
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -6 a + 6\) , \( 4 a + 16\bigr] \)
|
| 6912.3-j6
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -2 a - 10\) , \( -8 a - 8\bigr] \)
|
