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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([1,1]),K([-1,0]),K([0,0]),K([-5,665]),K([-4725,-4950])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 26000.6-b have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 4 & 3 & 2 & 4 & 12 & 6 & 12 \\
4 & 1 & 12 & 2 & 4 & 3 & 6 & 12 \\
3 & 12 & 1 & 6 & 12 & 4 & 2 & 4 \\
2 & 2 & 6 & 1 & 2 & 6 & 3 & 6 \\
4 & 4 & 12 & 2 & 1 & 12 & 6 & 3 \\
12 & 3 & 4 & 6 & 12 & 1 & 2 & 4 \\
6 & 6 & 2 & 3 & 6 & 2 & 1 & 2 \\
12 & 12 & 4 & 6 & 3 & 4 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 26000.6-b over \(\Q(\sqrt{-1}) \)
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sage:E.isogeny_class().curves
Isogeny class 26000.6-b contains
8 curves linked by isogenies of
degrees dividing 12.
| Curve label |
Weierstrass Coefficients |
| 26000.6-b1
| \( \bigl[i + 1\) , \( -1\) , \( 0\) , \( 665 i - 5\) , \( -4950 i - 4725\bigr] \)
|
| 26000.6-b2
| \( \bigl[0\) , \( i - 1\) , \( 0\) , \( -42 i + 56\) , \( 172 i + 179\bigr] \)
|
| 26000.6-b3
| \( \bigl[i + 1\) , \( -1\) , \( 0\) , \( -285 i + 145\) , \( -18130 i - 14965\bigr] \)
|
| 26000.6-b4
| \( \bigl[i + 1\) , \( -1\) , \( 0\) , \( 40 i - 5\) , \( -75 i - 100\bigr] \)
|
| 26000.6-b5
| \( \bigl[i + 1\) , \( -1\) , \( 0\) , \( -105 i + 135\) , \( -694 i - 567\bigr] \)
|
| 26000.6-b6
| \( \bigl[0\) , \( i - 1\) , \( 0\) , \( 158 i - 344\) , \( -308 i + 1039\bigr] \)
|
| 26000.6-b7
| \( \bigl[i + 1\) , \( -1\) , \( 0\) , \( -460 i + 745\) , \( -6375 i - 7500\bigr] \)
|
| 26000.6-b8
| \( \bigl[i + 1\) , \( -1\) , \( 0\) , \( -7355 i + 11885\) , \( -424194 i - 472567\bigr] \)
|
