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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([22, -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
\( x^{2} - x + 22 \); class number \(6\).
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sage:E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([-36,0]),K([-176,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 588.5-a have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 8 & 2 & 4 \\
8 & 1 & 2 & 4 & 4 & 8 \\
4 & 2 & 1 & 2 & 2 & 4 \\
8 & 4 & 2 & 1 & 4 & 8 \\
2 & 4 & 2 & 4 & 1 & 2 \\
4 & 8 & 4 & 8 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
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sage:E.isogeny_class().curves
Isogeny class 588.5-a contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 588.5-a1
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -36\) , \( -176\bigr] \)
|
| 588.5-a2
| \( \bigl[1\) , \( -1\) , \( 0\) , \( 3474\) , \( -31010\bigr] \)
|
| 588.5-a3
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -936\) , \( -3668\bigr] \)
|
| 588.5-a4
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -8226\) , \( 286474\bigr] \)
|
| 588.5-a5
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -756\) , \( -7808\bigr] \)
|
| 588.5-a6
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -12096\) , \( -509036\bigr] \)
|
