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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 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 \(i\), with minimal polynomial
\( x^{2} + 1 \); class number \(1\).
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sage:E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([59,-32]),K([155,177])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 26000.4-d have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 4 & 2 & 4 & 8 & 8 \\
4 & 1 & 2 & 4 & 8 & 8 \\
2 & 2 & 1 & 2 & 4 & 4 \\
4 & 4 & 2 & 1 & 2 & 2 \\
8 & 8 & 4 & 2 & 1 & 4 \\
8 & 8 & 4 & 2 & 4 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 26000.4-d over \(\Q(\sqrt{-1}) \)
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sage:E.isogeny_class().curves
Isogeny class 26000.4-d contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 26000.4-d1
| \( \bigl[0\) , \( i\) , \( 0\) , \( -32 i + 59\) , \( 177 i + 155\bigr] \)
|
| 26000.4-d2
| \( \bigl[i + 1\) , \( i - 1\) , \( i + 1\) , \( -79 i - 204\) , \( 570 i + 1169\bigr] \)
|
| 26000.4-d3
| \( \bigl[i + 1\) , \( i - 1\) , \( i + 1\) , \( i - 19\) , \( 21 i + 1\bigr] \)
|
| 26000.4-d4
| \( \bigl[i + 1\) , \( i - 1\) , \( i + 1\) , \( i + 106\) , \( 346 i - 99\bigr] \)
|
| 26000.4-d5
| \( \bigl[i + 1\) , \( i - 1\) , \( i + 1\) , \( -99 i + 656\) , \( -5844 i - 1679\bigr] \)
|
| 26000.4-d6
| \( \bigl[i + 1\) , \( i - 1\) , \( i + 1\) , \( 101 i + 1556\) , \( 24336 i - 2919\bigr] \)
|
