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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -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
\( x^{2} - x + 2 \); class number \(1\).
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sage:E = EllipticCurve([K([1,1]),K([1,-1]),K([0,0]),K([-448,-149]),K([-3504,-1964])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 784.4-a have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 2 & 4 & 8 & 4 \\
8 & 1 & 4 & 8 & 4 & 2 \\
2 & 4 & 1 & 2 & 4 & 2 \\
4 & 8 & 2 & 1 & 8 & 4 \\
8 & 4 & 4 & 8 & 1 & 2 \\
4 & 2 & 2 & 4 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 784.4-a over \(\Q(\sqrt{-7}) \)
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sage:E.isogeny_class().curves
Isogeny class 784.4-a contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 784.4-a1
| \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -149 a - 448\) , \( -1964 a - 3504\bigr] \)
|
| 784.4-a2
| \( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -14 a + 96\) , \( 287 a + 36\bigr] \)
|
| 784.4-a3
| \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -9 a - 28\) , \( -32 a - 60\bigr] \)
|
| 784.4-a4
| \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 3 a + 12\) , \( 26 a - 40\bigr] \)
|
| 784.4-a5
| \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -54 a + 66\) , \( 120 a + 304\bigr] \)
|
| 784.4-a6
| \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 16 a - 4\) , \( 8 a + 24\bigr] \)
|
