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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 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 \(a\), with minimal polynomial
\( x^{2} - 3 \); class number \(1\).
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sage:E = EllipticCurve([K([1,1]),K([1,0]),K([0,0]),K([-9,3]),K([-27,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 96.1-a have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 8 & 2 & 4 \\
8 & 1 & 2 & 4 & 4 & 8 \\
4 & 2 & 1 & 2 & 2 & 4 \\
8 & 4 & 2 & 1 & 4 & 8 \\
2 & 4 & 2 & 4 & 1 & 2 \\
4 & 8 & 4 & 8 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 96.1-a over \(\Q(\sqrt{3}) \)
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sage:E.isogeny_class().curves
Isogeny class 96.1-a contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 96.1-a1
| \( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 3 a - 9\) , \( -27\bigr] \)
|
| 96.1-a2
| \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 50243 a - 87023\) , \( -8019946 a + 13890954\bigr] \)
|
| 96.1-a3
| \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 920 a - 1591\) , \( -18385 a + 31843\bigr] \)
|
| 96.1-a4
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -3102 a + 5373\) , \( -535777 a + 927993\bigr] \)
|
| 96.1-a5
| \( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 56 a - 90\) , \( 222 a - 381\bigr] \)
|
| 96.1-a6
| \( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -109 a + 195\) , \( 1392 a - 2409\bigr] \)
|
