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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([0,1]),K([0,0]),K([-7038714,4063804]),K([-10114941022,5839863922])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 288.1-c 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 288.1-c over \(\Q(\sqrt{3}) \)
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sage:E.isogeny_class().curves
Isogeny class 288.1-c contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 288.1-c1
| \( \bigl[0\) , \( a\) , \( 0\) , \( 4063804 a - 7038714\) , \( 5839863922 a - 10114941022\bigr] \)
|
| 288.1-c2
| \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 381758 a - 661224\) , \( -168866036 a + 292484554\bigr] \)
|
| 288.1-c3
| \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 23873 a - 41349\) , \( -2630312 a + 4555834\bigr] \)
|
| 288.1-c4
| \( \bigl[0\) , \( a\) , \( 0\) , \( 113682 a - 196902\) , \( -3469554 a + 6009444\bigr] \)
|
| 288.1-c5
| \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 77 a - 140\) , \( -1664 a + 2879\bigr] \)
|
| 288.1-c6
| \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -1569806 a + 2718984\) , \( -176681708 a + 306021695\bigr] \)
|
