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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -3, -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
\( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).
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sage:E = EllipticCurve([K([-1,0,1]),K([1,0,-1]),K([1,1,0]),K([-2052,5213,4455]),K([16900486,-42913527,-36675517])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 400.2-a have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 2 & 8 & 4 & 8 & 4 \\
2 & 1 & 4 & 2 & 4 & 2 \\
8 & 4 & 1 & 2 & 4 & 8 \\
4 & 2 & 2 & 1 & 2 & 4 \\
8 & 4 & 4 & 2 & 1 & 8 \\
4 & 2 & 8 & 4 & 8 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 400.2-a over 3.3.148.1
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sage:E.isogeny_class().curves
Isogeny class 400.2-a contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 400.2-a1
| \( \bigl[a^{2} - 1\) , \( -a^{2} + 1\) , \( a + 1\) , \( 4455 a^{2} + 5213 a - 2052\) , \( -36675517 a^{2} - 42913527 a + 16900486\bigr] \)
|
| 400.2-a2
| \( \bigl[a^{2} - 1\) , \( -a^{2} + 2 a + 2\) , \( 0\) , \( 2 a^{2} - 3\) , \( 0\bigr] \)
|
| 400.2-a3
| \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( 0\) , \( 42 a^{2} + 21 a - 63\) , \( 57 a^{2} + 130 a + 66\bigr] \)
|
| 400.2-a4
| \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( 0\) , \( -8 a^{2} - 9 a + 2\) , \( 13 a^{2} + 16 a - 6\bigr] \)
|
| 400.2-a5
| \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( 0\) , \( -118 a^{2} - 119 a + 47\) , \( 1199 a^{2} + 1448 a - 566\bigr] \)
|
| 400.2-a6
| \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a + 1\) , \( -61 a^{2} - 70 a + 29\) , \( -459 a^{2} - 537 a + 212\bigr] \)
|
