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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([31, -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
\( x^{2} - x + 31 \); class number \(2\).
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sage:E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-110,0]),K([-880,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 75.1-b have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 16 & 8 & 4 & 8 & 2 & 16 & 4 \\
16 & 1 & 8 & 4 & 2 & 8 & 4 & 16 \\
8 & 8 & 1 & 2 & 4 & 4 & 8 & 8 \\
4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\
8 & 2 & 4 & 2 & 1 & 4 & 2 & 8 \\
2 & 8 & 4 & 2 & 4 & 1 & 8 & 2 \\
16 & 4 & 8 & 4 & 2 & 8 & 1 & 16 \\
4 & 16 & 8 & 4 & 8 & 2 & 16 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
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sage:E.isogeny_class().curves
Isogeny class 75.1-b contains
8 curves linked by isogenies of
degrees dividing 16.
| Curve label |
Weierstrass Coefficients |
| 75.1-b1
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \)
|
| 75.1-b2
| \( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \)
|
| 75.1-b3
| \( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \)
|
| 75.1-b4
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \)
|
| 75.1-b5
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \)
|
| 75.1-b6
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \)
|
| 75.1-b7
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \)
|
| 75.1-b8
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \)
|
