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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, -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
\( x^{2} - x + 4 \); class number \(2\).
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sage:E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([-460,115]),K([-3808,1628])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 800.2-b have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 2 & 2 & 3 & 6 & 6 & 6 & 2 \\
2 & 1 & 4 & 6 & 12 & 12 & 3 & 4 \\
2 & 4 & 1 & 6 & 12 & 3 & 12 & 4 \\
3 & 6 & 6 & 1 & 2 & 2 & 2 & 6 \\
6 & 12 & 12 & 2 & 1 & 4 & 4 & 3 \\
6 & 12 & 3 & 2 & 4 & 1 & 4 & 12 \\
6 & 3 & 12 & 2 & 4 & 4 & 1 & 12 \\
2 & 4 & 4 & 6 & 3 & 12 & 12 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
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sage:E.isogeny_class().curves
Isogeny class 800.2-b contains
8 curves linked by isogenies of
degrees dividing 12.
| Curve label |
Weierstrass Coefficients |
| 800.2-b1
| \( \bigl[a\) , \( -a + 1\) , \( a\) , \( 115 a - 460\) , \( 1628 a - 3808\bigr] \)
|
| 800.2-b2
| \( \bigl[a\) , \( a\) , \( 0\) , \( 20 a - 3\) , \( 35 a + 62\bigr] \)
|
| 800.2-b3
| \( \bigl[a\) , \( a\) , \( 0\) , \( -15 a - 23\) , \( -70 a + 2\bigr] \)
|
| 800.2-b4
| \( \bigl[a\) , \( -1\) , \( 0\) , \( -10 a + 40\) , \( 0\bigr] \)
|
| 800.2-b5
| \( \bigl[a\) , \( -1\) , \( 0\) , \( 40 a - 160\) , \( -110 a + 280\bigr] \)
|
| 800.2-b6
| \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -79 a + 444\) , \( -1436 a - 780\bigr] \)
|
| 800.2-b7
| \( \bigl[a\) , \( -1\) , \( 0\) , \( -135 a + 415\) , \( 1125 a + 2125\bigr] \)
|
| 800.2-b8
| \( \bigl[a\) , \( -a + 1\) , \( a\) , \( 1915 a - 7660\) , \( 93068 a - 231328\bigr] \)
|
