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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -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
\( x^{2} - x + 1 \); class number \(1\).
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sage:E = EllipticCurve([K([1,0]),K([-1,0]),K([0,1]),K([12,-26]),K([-50,161])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 441.2-a have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 12 & 4 & 12 & 6 & 2 & 3 & 4 \\
12 & 1 & 12 & 4 & 2 & 6 & 4 & 3 \\
4 & 12 & 1 & 3 & 6 & 2 & 12 & 4 \\
12 & 4 & 3 & 1 & 2 & 6 & 4 & 12 \\
6 & 2 & 6 & 2 & 1 & 3 & 2 & 6 \\
2 & 6 & 2 & 6 & 3 & 1 & 6 & 2 \\
3 & 4 & 12 & 4 & 2 & 6 & 1 & 12 \\
4 & 3 & 4 & 12 & 6 & 2 & 12 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 441.2-a over \(\Q(\sqrt{-3}) \)
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sage:E.isogeny_class().curves
Isogeny class 441.2-a contains
8 curves linked by isogenies of
degrees dividing 12.
| Curve label |
Weierstrass Coefficients |
| 441.2-a1
| \( \bigl[1\) , \( -1\) , \( a\) , \( -26 a + 12\) , \( 161 a - 50\bigr] \)
|
| 441.2-a2
| \( \bigl[1\) , \( -1\) , \( a + 1\) , \( 25 a - 14\) , \( -162 a + 111\bigr] \)
|
| 441.2-a3
| \( \bigl[1\) , \( -1\) , \( a\) , \( 4 a - 3\) , \( -4 a + 1\bigr] \)
|
| 441.2-a4
| \( \bigl[1\) , \( -1\) , \( a + 1\) , \( -5 a + 1\) , \( 3 a - 3\bigr] \)
|
| 441.2-a5
| \( \bigl[1\) , \( -1\) , \( a + 1\) , \( -20 a + 1\) , \( -30 a + 18\bigr] \)
|
| 441.2-a6
| \( \bigl[1\) , \( -1\) , \( a\) , \( 19 a - 18\) , \( 29 a - 11\bigr] \)
|
| 441.2-a7
| \( \bigl[1\) , \( -1\) , \( a + 1\) , \( -305 a + 16\) , \( -2190 a + 1269\bigr] \)
|
| 441.2-a8
| \( \bigl[1\) , \( -1\) , \( a\) , \( 304 a - 288\) , \( 2189 a - 920\bigr] \)
|
