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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([0,0]),K([-1,0]),K([0,0]),K([-544,0]),K([13888,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 16128.5-i have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 8 & 4 & 16 & 8 & 16 & 2 & 4 \\
8 & 1 & 2 & 2 & 4 & 2 & 4 & 8 \\
4 & 2 & 1 & 4 & 2 & 4 & 2 & 4 \\
16 & 2 & 4 & 1 & 8 & 4 & 8 & 16 \\
8 & 4 & 2 & 8 & 1 & 8 & 4 & 8 \\
16 & 2 & 4 & 4 & 8 & 1 & 8 & 16 \\
2 & 4 & 2 & 8 & 4 & 8 & 1 & 2 \\
4 & 8 & 4 & 16 & 8 & 16 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 16128.5-i over \(\Q(\sqrt{-7}) \)
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sage:E.isogeny_class().curves
Isogeny class 16128.5-i contains
8 curves linked by isogenies of
degrees dividing 16.
| Curve label |
Weierstrass Coefficients |
| 16128.5-i1
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -544\) , \( 13888\bigr] \)
|
| 16128.5-i2
| \( \bigl[0\) , \( -1\) , \( 0\) , \( 16\) , \( 0\bigr] \)
|
| 16128.5-i3
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -64\) , \( 64\bigr] \)
|
| 16128.5-i4
| \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -31 a + 26\) , \( -22 a + 88\bigr] \)
|
| 16128.5-i5
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -624\) , \( -5760\bigr] \)
|
| 16128.5-i6
| \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 33 a - 6\) , \( 54 a + 60\bigr] \)
|
| 16128.5-i7
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -784\) , \( 8704\bigr] \)
|
| 16128.5-i8
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -12544\) , \( 544960\bigr] \)
|
