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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([1,0]),K([7,-7]),K([5,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 8281.5-a have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrr}
1 & 9 & 3 & 9 & 9 \\
9 & 1 & 3 & 9 & 9 \\
3 & 3 & 1 & 3 & 3 \\
9 & 9 & 3 & 1 & 9 \\
9 & 9 & 3 & 9 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 8281.5-a over \(\Q(\sqrt{-3}) \)
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sage:E.isogeny_class().curves
Isogeny class 8281.5-a contains
5 curves linked by isogenies of
degrees dividing 9.
| Curve label |
Weierstrass Coefficients |
| 8281.5-a1
| \( \bigl[0\) , \( -a\) , \( 1\) , \( -7 a + 7\) , \( 5\bigr] \)
|
| 8281.5-a2
| \( \bigl[0\) , \( -a\) , \( 1\) , \( -117 a + 117\) , \( -1245\bigr] \)
|
| 8281.5-a3
| \( \bigl[0\) , \( -a\) , \( 1\) , \( 13 a - 13\) , \( 42\bigr] \)
|
| 8281.5-a4
| \( \bigl[0\) , \( -a\) , \( 1\) , \( 503 a - 1273\) , \( -9086 a + 16555\bigr] \)
|
| 8281.5-a5
| \( \bigl[0\) , \( -a\) , \( 1\) , \( 1273 a - 503\) , \( 9086 a + 7469\bigr] \)
|
