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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([0,0]),K([-1,-1]),K([0,0]),K([-2346237,1354602]),K([-1947975909,1124664415])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 768.1-p have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 2 & 8 & 4 \\
8 & 1 & 2 & 4 & 4 & 8 \\
4 & 2 & 1 & 2 & 2 & 4 \\
2 & 4 & 2 & 1 & 4 & 2 \\
8 & 4 & 2 & 4 & 1 & 8 \\
4 & 8 & 4 & 2 & 8 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 768.1-p over \(\Q(\sqrt{3}) \)
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sage:E.isogeny_class().curves
Isogeny class 768.1-p contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 768.1-p1
| \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 1354602 a - 2346237\) , \( 1124664415 a - 1947975909\bigr] \)
|
| 768.1-p2
| \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 36546 a - 63297\) , \( -4995933 a + 8653209\bigr] \)
|
| 768.1-p3
| \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 2286 a - 3957\) , \( -77553 a + 134325\bigr] \)
|
| 768.1-p4
| \( \bigl[0\) , \( -1\) , \( 0\) , \( 37894 a - 65634\) , \( -667716 a + 1156518\bigr] \)
|
| 768.1-p5
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -10 a - 41\) , \( -11 a + 177\bigr] \)
|
| 768.1-p6
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -150276 a + 260286\) , \( -5169562 a + 8953944\bigr] \)
|
