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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([0,1]),K([1,0]),K([0,1]),K([0,0]),K([14,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 289.2-a have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 4 & 2 & 2 & 2 & 4 \\
4 & 1 & 8 & 8 & 2 & 4 \\
2 & 8 & 1 & 4 & 4 & 8 \\
2 & 8 & 4 & 1 & 4 & 8 \\
2 & 2 & 4 & 4 & 1 & 2 \\
4 & 4 & 8 & 8 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 289.2-a over \(\Q(\sqrt{-1}) \)
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sage:E.isogeny_class().curves
Isogeny class 289.2-a contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 289.2-a1
| \( \bigl[i\) , \( 1\) , \( i\) , \( 0\) , \( 14\bigr] \)
|
| 289.2-a2
| \( \bigl[i\) , \( 1\) , \( i\) , \( 0\) , \( 0\bigr] \)
|
| 289.2-a3
| \( \bigl[i\) , \( 1\) , \( i\) , \( -75 i + 40\) , \( -38 i + 304\bigr] \)
|
| 289.2-a4
| \( \bigl[i\) , \( 1\) , \( i\) , \( 75 i + 40\) , \( 38 i + 304\bigr] \)
|
| 289.2-a5
| \( \bigl[1\) , \( -1\) , \( 1\) , \( -6\) , \( -4\bigr] \)
|
| 289.2-a6
| \( \bigl[1\) , \( -1\) , \( 1\) , \( -91\) , \( -310\bigr] \)
|
