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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([49, -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
\( x^{2} - x + 49 \); class number \(4\).
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sage:E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-338,0]),K([-7969,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 60.1-b have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 3 & 4 & 12 & 6 & 2 & 12 & 4 \\
3 & 1 & 12 & 4 & 2 & 6 & 4 & 12 \\
4 & 12 & 1 & 12 & 6 & 2 & 3 & 4 \\
12 & 4 & 12 & 1 & 2 & 6 & 4 & 3 \\
6 & 2 & 6 & 2 & 1 & 3 & 2 & 6 \\
2 & 6 & 2 & 6 & 3 & 1 & 6 & 2 \\
12 & 4 & 3 & 4 & 2 & 6 & 1 & 12 \\
4 & 12 & 4 & 3 & 6 & 2 & 12 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
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sage:E.isogeny_class().curves
Isogeny class 60.1-b contains
8 curves linked by isogenies of
degrees dividing 12.
| Curve label |
Weierstrass Coefficients |
| 60.1-b1
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -338\) , \( -7969\bigr] \)
|
| 60.1-b2
| \( \bigl[1\) , \( 1\) , \( 1\) , \( 37\) , \( 281\bigr] \)
|
| 60.1-b3
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -11338\) , \( -67969\bigr] \)
|
| 60.1-b4
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -1713\) , \( -24219\bigr] \)
|
| 60.1-b5
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -463\) , \( 3281\bigr] \)
|
| 60.1-b6
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -8338\) , \( -295969\bigr] \)
|
| 60.1-b7
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -7213\) , \( 232781\bigr] \)
|
| 60.1-b8
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -133338\) , \( -18795969\bigr] \)
|
