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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([-98036652,56601488]),K([526050184174,-303715215440])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 2304.1-x have
rank \( 0 \).
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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 2304.1-x over \(\Q(\sqrt{3}) \)
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sage:E.isogeny_class().curves
Isogeny class 2304.1-x contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 2304.1-x1
| \( \bigl[0\) , \( a\) , \( 0\) , \( 56601488 a - 98036652\) , \( -303715215440 a + 526050184174\bigr] \)
|
| 2304.1-x2
| \( \bigl[0\) , \( a\) , \( 0\) , \( 1527032 a - 2644896\) , \( 1350928288 a - 2339876432\bigr] \)
|
| 2304.1-x3
| \( \bigl[0\) , \( a\) , \( 0\) , \( 95492 a - 165396\) , \( 21042496 a - 36446672\bigr] \)
|
| 2304.1-x4
| \( \bigl[0\) , \( a\) , \( 0\) , \( 1583386 a - 2742504\) , \( 176024176 a - 304882816\bigr] \)
|
| 2304.1-x5
| \( \bigl[0\) , \( a\) , \( 0\) , \( 312 a - 552\) , \( 13308 a - 23040\bigr] \)
|
| 2304.1-x6
| \( \bigl[0\) , \( a\) , \( 0\) , \( -6279224 a + 10875936\) , \( 1413453664 a - 2448173560\bigr] \)
|
