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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -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
\( x^{2} - x - 5 \); class number \(1\).
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sage:E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-34,0]),K([-217,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 21.1-b have
rank \( 1 \).
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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)
Elliptic curves in class 21.1-b over \(\Q(\sqrt{21}) \)
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sage:E.isogeny_class().curves
Isogeny class 21.1-b contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 21.1-b1
| \( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \)
|
| 21.1-b2
| \( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \)
|
| 21.1-b3
| \( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \)
|
| 21.1-b4
| \( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \)
|
| 21.1-b5
| \( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \)
|
| 21.1-b6
| \( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \)
|
