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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([114, -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
\( x^{2} - x + 114 \); class number \(20\).
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sage:E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-4263,0]),K([-109219,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The rank \(r\) of the
elliptic curves in class 28.2-d satisfy
\(0 \le r \le 1\).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 9 & 3 & 6 & 18 & 2 \\
9 & 1 & 3 & 6 & 2 & 18 \\
3 & 3 & 1 & 2 & 6 & 6 \\
6 & 6 & 2 & 1 & 3 & 3 \\
18 & 2 & 6 & 3 & 1 & 9 \\
2 & 18 & 6 & 3 & 9 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
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sage:E.isogeny_class().curves
Isogeny class 28.2-d contains
6 curves linked by isogenies of
degrees dividing 18.
| Curve label |
Weierstrass Coefficients |
| 28.2-d1
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -4263\) , \( -109219\bigr] \)
|
| 28.2-d2
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -13\) , \( 31\bigr] \)
|
| 28.2-d3
| \( \bigl[1\) , \( 1\) , \( 1\) , \( 112\) , \( -719\bigr] \)
|
| 28.2-d4
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -888\) , \( -8719\bigr] \)
|
| 28.2-d5
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -263\) , \( 1531\bigr] \)
|
| 28.2-d6
| \( \bigl[1\) , \( 1\) , \( 1\) , \( -68263\) , \( -6893219\bigr] \)
|
