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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([-85,470]),K([-5005,-1671])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 7168.5-f have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 2 & 4 & 8 & 4 \\
8 & 1 & 4 & 8 & 4 & 2 \\
2 & 4 & 1 & 2 & 4 & 2 \\
4 & 8 & 2 & 1 & 8 & 4 \\
8 & 4 & 4 & 8 & 1 & 2 \\
4 & 2 & 2 & 4 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 7168.5-f over \(\Q(\sqrt{-7}) \)
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sage:E.isogeny_class().curves
Isogeny class 7168.5-f contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 7168.5-f1
| \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 470 a - 85\) , \( -1671 a - 5005\bigr] \)
|
| 7168.5-f2
| \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -53 a - 51\) , \( 336 a + 23\bigr] \)
|
| 7168.5-f3
| \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 30 a - 5\) , \( -15 a - 93\bigr] \)
|
| 7168.5-f4
| \( \bigl[0\) , \( 1\) , \( 0\) , \( -11 a + 1\) , \( 33 a - 39\bigr] \)
|
| 7168.5-f5
| \( \bigl[0\) , \( -1\) , \( 0\) , \( 5 a - 95\) , \( 99 a + 367\bigr] \)
|
| 7168.5-f6
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -15 a + 25\) , \( 3 a + 47\bigr] \)
|
