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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([0,0]),K([1,1]),K([165,-92]),K([-668,-698])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 800.1-a have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 2 & 4 & 8 \\
8 & 1 & 2 & 4 & 8 & 4 \\
4 & 2 & 1 & 2 & 4 & 2 \\
2 & 4 & 2 & 1 & 2 & 4 \\
4 & 8 & 4 & 2 & 1 & 8 \\
8 & 4 & 2 & 4 & 8 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 800.1-a over \(\Q(\sqrt{-1}) \)
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sage:E.isogeny_class().curves
Isogeny class 800.1-a contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 800.1-a1
| \( \bigl[i + 1\) , \( 0\) , \( i + 1\) , \( -92 i + 165\) , \( -698 i - 668\bigr] \)
|
| 800.1-a2
| \( \bigl[0\) , \( -i\) , \( 0\) , \( 4 i - 14\) , \( 14 i - 18\bigr] \)
|
| 800.1-a3
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -6 i\) , \( 12 i - 4\bigr] \)
|
| 800.1-a4
| \( \bigl[i + 1\) , \( -i\) , \( i + 1\) , \( -7 i + 10\) , \( 10 i + 9\bigr] \)
|
| 800.1-a5
| \( \bigl[i + 1\) , \( -i\) , \( i + 1\) , \( -42 i + 15\) , \( 43 i - 110\bigr] \)
|
| 800.1-a6
| \( \bigl[i + 1\) , \( 0\) , \( 0\) , \( 29 i + 5\) , \( -30 i - 58\bigr] \)
|
