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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([-2,-1,1]),K([0,0,0]),K([-2,-1,1]),K([-1,1,-12]),K([-18,50,14])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 200.2-d have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 8 & 4 & 2 & 4 \\
8 & 1 & 4 & 2 & 4 & 8 \\
8 & 4 & 1 & 2 & 4 & 8 \\
4 & 2 & 2 & 1 & 2 & 4 \\
2 & 4 & 4 & 2 & 1 & 2 \\
4 & 8 & 8 & 4 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 200.2-d over 3.3.148.1
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sage:E.isogeny_class().curves
Isogeny class 200.2-d contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 200.2-d1
| \( \bigl[a^{2} - a - 2\) , \( 0\) , \( a^{2} - a - 2\) , \( -12 a^{2} + a - 1\) , \( 14 a^{2} + 50 a - 18\bigr] \)
|
| 200.2-d2
| \( \bigl[a^{2} - a - 2\) , \( 0\) , \( 0\) , \( -160 a^{2} + 341 a - 91\) , \( 1620 a^{2} - 4242 a + 1163\bigr] \)
|
| 200.2-d3
| \( \bigl[a^{2} - a - 2\) , \( 0\) , \( 0\) , \( 21 a - 21\) , \( 22 a^{2} - 80 a + 39\bigr] \)
|
| 200.2-d4
| \( \bigl[a^{2} - a - 2\) , \( 0\) , \( 0\) , \( -10 a^{2} + 21 a - 6\) , \( 24 a^{2} - 62 a + 17\bigr] \)
|
| 200.2-d5
| \( \bigl[a^{2} - 1\) , \( a^{2} - 3\) , \( 0\) , \( -4 a^{2} + 14 a - 3\) , \( 8 a^{2} - 18 a + 6\bigr] \)
|
| 200.2-d6
| \( \bigl[a^{2} - 1\) , \( a + 1\) , \( a + 1\) , \( 608 a^{2} + 712 a - 279\) , \( 1845556 a^{2} + 2159460 a - 850453\bigr] \)
|
