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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([0,1]),K([0,0]),K([3337,1018]),K([342660,104632])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 171.1-e 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-e over \(\Q(\sqrt{57}) \)
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sage:E.isogeny_class().curves
Isogeny class 171.1-e contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 171.1-e1
| \( \bigl[1\) , \( a\) , \( 0\) , \( 1018 a + 3337\) , \( 104632 a + 342660\bigr] \)
|
| 171.1-e2
| \( \bigl[1\) , \( a\) , \( 0\) , \( 6 a - 18\) , \( -18 a + 81\bigr] \)
|
| 171.1-e3
| \( \bigl[1\) , \( a\) , \( 0\) , \( -182 a - 593\) , \( -2528 a - 8280\bigr] \)
|
| 171.1-e4
| \( \bigl[1\) , \( a\) , \( 0\) , \( -782 a - 2558\) , \( 19744 a + 64659\bigr] \)
|
| 171.1-e5
| \( \bigl[1\) , \( a\) , \( 0\) , \( -9960 a + 42585\) , \( -2264742 a + 9681588\bigr] \)
|
| 171.1-e6
| \( \bigl[1\) , \( a\) , \( 0\) , \( -12182 a - 39893\) , \( 1403704 a + 4597014\bigr] \)
|
