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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -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
\( x^{2} - x - 14 \); class number \(1\).
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sage:E = EllipticCurve([K([1,0]),K([1,-1]),K([0,0]),K([4355,-1018]),K([447292,-104632])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 171.1-h have
rank \( 0 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 2 & 8 & 4 \\
8 & 1 & 2 & 4 & 4 & 8 \\
4 & 2 & 1 & 2 & 2 & 4 \\
2 & 4 & 2 & 1 & 4 & 2 \\
8 & 4 & 2 & 4 & 1 & 8 \\
4 & 8 & 4 & 2 & 8 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 171.1-h over \(\Q(\sqrt{57}) \)
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sage:E.isogeny_class().curves
Isogeny class 171.1-h contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 171.1-h1
| \( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -1018 a + 4355\) , \( -104632 a + 447292\bigr] \)
|
| 171.1-h2
| \( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 9960 a + 32625\) , \( 2264742 a + 7416846\bigr] \)
|
| 171.1-h3
| \( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 182 a - 775\) , \( 2528 a - 10808\bigr] \)
|
| 171.1-h4
| \( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 782 a - 3340\) , \( -19744 a + 84403\bigr] \)
|
| 171.1-h5
| \( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -6 a - 12\) , \( 18 a + 63\bigr] \)
|
| 171.1-h6
| \( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 12182 a - 52075\) , \( -1403704 a + 6000718\bigr] \)
|
