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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -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
\( x^{2} - x - 14 \); class number \(1\).
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sage:E = EllipticCurve([K([1,0]),K([0,-1]),K([0,1]),K([-6745970,-2059892]),K([10278231250,3138470553])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 196.1-e have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 9 & 3 & 6 & 18 & 2 \\
9 & 1 & 3 & 6 & 2 & 18 \\
3 & 3 & 1 & 2 & 6 & 6 \\
6 & 6 & 2 & 1 & 3 & 3 \\
18 & 2 & 6 & 3 & 1 & 9 \\
2 & 18 & 6 & 3 & 9 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 196.1-e over \(\Q(\sqrt{57}) \)
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sage:E.isogeny_class().curves
Isogeny class 196.1-e contains
6 curves linked by isogenies of
degrees dividing 18.
| Curve label |
Weierstrass Coefficients |
| 196.1-e1
| \( \bigl[1\) , \( -a\) , \( a\) , \( -2059892 a - 6745970\) , \( 3138470553 a + 10278231250\bigr] \)
|
| 196.1-e2
| \( \bigl[1\) , \( -a\) , \( a\) , \( -6292 a - 20600\) , \( -1059817 a - 3470814\bigr] \)
|
| 196.1-e3
| \( \bigl[1\) , \( -a\) , \( a\) , \( 54108 a + 177205\) , \( 22267918 a + 72925587\bigr] \)
|
| 196.1-e4
| \( \bigl[1\) , \( -a\) , \( a\) , \( -429092 a - 1405235\) , \( 244235478 a + 799850971\bigr] \)
|
| 196.1-e5
| \( \bigl[1\) , \( -a\) , \( a\) , \( -127092 a - 416210\) , \( -47715287 a - 156263616\bigr] \)
|
| 196.1-e6
| \( \bigl[1\) , \( -a\) , \( a\) , \( -32984692 a - 108022130\) , \( 200392463513 a + 656268729042\bigr] \)
|
