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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([0,-1]),K([0,0]),K([1022,-853]),K([17334,-809])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 28672.7-w have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 2 & 8 & 4 & 4 \\
8 & 1 & 4 & 4 & 8 & 2 \\
2 & 4 & 1 & 4 & 2 & 2 \\
8 & 4 & 4 & 1 & 8 & 2 \\
4 & 8 & 2 & 8 & 1 & 4 \\
4 & 2 & 2 & 2 & 4 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 28672.7-w over \(\Q(\sqrt{-7}) \)
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sage:E.isogeny_class().curves
Isogeny class 28672.7-w contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 28672.7-w1
| \( \bigl[0\) , \( -a\) , \( 0\) , \( -853 a + 1022\) , \( -809 a + 17334\bigr] \)
|
| 28672.7-w2
| \( \bigl[0\) , \( a\) , \( 0\) , \( 159 a - 58\) , \( -491 a - 798\bigr] \)
|
| 28672.7-w3
| \( \bigl[0\) , \( -a\) , \( 0\) , \( -53 a + 62\) , \( -9 a + 246\bigr] \)
|
| 28672.7-w4
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 85 a + 105\) , \( 169 a - 1299\bigr] \)
|
| 28672.7-w5
| \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 21 a - 23\) , \( -105 a + 51\bigr] \)
|
| 28672.7-w6
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 5 a - 55\) , \( 41 a - 147\bigr] \)
|
