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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([0,0]),K([0,1]),K([0,0]),K([-6,11]),K([-1,11])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 192.1-a have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 4 & 16 & 2 & 4 & 8 & 8 & 16 \\
4 & 1 & 16 & 2 & 4 & 8 & 8 & 16 \\
16 & 16 & 1 & 8 & 4 & 2 & 8 & 4 \\
2 & 2 & 8 & 1 & 2 & 4 & 4 & 8 \\
4 & 4 & 4 & 2 & 1 & 2 & 2 & 4 \\
8 & 8 & 2 & 4 & 2 & 1 & 4 & 2 \\
8 & 8 & 8 & 4 & 2 & 4 & 1 & 8 \\
16 & 16 & 4 & 8 & 4 & 2 & 8 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 192.1-a over \(\Q(\sqrt{-3}) \)
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sage:E.isogeny_class().curves
Isogeny class 192.1-a contains
8 curves linked by isogenies of
degrees dividing 16.
| Curve label |
Weierstrass Coefficients |
| 192.1-a1
| \( \bigl[0\) , \( a\) , \( 0\) , \( 11 a - 6\) , \( 11 a - 1\bigr] \)
|
| 192.1-a2
| \( \bigl[0\) , \( a\) , \( 0\) , \( 6 a - 11\) , \( -11 a + 10\bigr] \)
|
| 192.1-a3
| \( \bigl[0\) , \( a\) , \( 0\) , \( 16 a - 16\) , \( -180\bigr] \)
|
| 192.1-a4
| \( \bigl[0\) , \( a\) , \( 0\) , \( a - 1\) , \( 0\bigr] \)
|
| 192.1-a5
| \( \bigl[0\) , \( a\) , \( 0\) , \( -4 a + 4\) , \( 4\bigr] \)
|
| 192.1-a6
| \( \bigl[0\) , \( a\) , \( 0\) , \( -24 a + 24\) , \( -36\bigr] \)
|
| 192.1-a7
| \( \bigl[0\) , \( a\) , \( 0\) , \( -64 a + 64\) , \( 220\bigr] \)
|
| 192.1-a8
| \( \bigl[0\) , \( a\) , \( 0\) , \( -384 a + 384\) , \( -2772\bigr] \)
|
