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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([182, 0, 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
\( x^{2} + 182 \); class number \(12\).
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sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([69,0]),K([23,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 14.1-d have
rank \( 1 \).
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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 14.1-d contains
6 curves linked by isogenies of
degrees dividing 18.
| Curve label |
Weierstrass Coefficients |
| 14.1-d1
| \( \bigl[a\) , \( 1\) , \( a\) , \( 69\) , \( 23\bigr] \)
|
| 14.1-d2
| \( \bigl[a\) , \( 1\) , \( a\) , \( 749\) , \( -3185\bigr] \)
|
| 14.1-d3
| \( \bigl[a\) , \( 1\) , \( a\) , \( 769\) , \( -3533\bigr] \)
|
| 14.1-d4
| \( \bigl[a\) , \( 1\) , \( a\) , \( 609\) , \( -1645\bigr] \)
|
| 14.1-d5
| \( \bigl[a\) , \( 1\) , \( a\) , \( 709\) , \( -2489\bigr] \)
|
| 14.1-d6
| \( \bigl[a\) , \( 1\) , \( a\) , \( -10171\) , \( -280553\bigr] \)
|
