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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([1,0]),K([-1,0]),K([0,0]),K([-306,0]),K([5859,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 567.1-a have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 8 & 4 & 8 & 16 & 16 & 2 & 4 \\
8 & 1 & 2 & 4 & 2 & 2 & 4 & 8 \\
4 & 2 & 1 & 2 & 4 & 4 & 2 & 4 \\
8 & 4 & 2 & 1 & 8 & 8 & 4 & 8 \\
16 & 2 & 4 & 8 & 1 & 4 & 8 & 16 \\
16 & 2 & 4 & 8 & 4 & 1 & 8 & 16 \\
2 & 4 & 2 & 4 & 8 & 8 & 1 & 2 \\
4 & 8 & 4 & 8 & 16 & 16 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 567.1-a over \(\Q(\sqrt{-7}) \)
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sage:E.isogeny_class().curves
Isogeny class 567.1-a contains
8 curves linked by isogenies of
degrees dividing 16.
| Curve label |
Weierstrass Coefficients |
| 567.1-a1
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -306\) , \( 5859\bigr] \)
|
| 567.1-a2
| \( \bigl[1\) , \( -1\) , \( 0\) , \( 9\) , \( 0\bigr] \)
|
| 567.1-a3
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -36\) , \( 27\bigr] \)
|
| 567.1-a4
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -351\) , \( -2430\bigr] \)
|
| 567.1-a5
| \( \bigl[1\) , \( -1\) , \( 0\) , \( 18 a - 3\) , \( 10 a + 36\bigr] \)
|
| 567.1-a6
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -18 a + 15\) , \( -10 a + 46\bigr] \)
|
| 567.1-a7
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -441\) , \( 3672\bigr] \)
|
| 567.1-a8
| \( \bigl[1\) , \( -1\) , \( 0\) , \( -7056\) , \( 229905\bigr] \)
|
