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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 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
\( x^{2} - 14 \); class number \(1\).
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sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-33003,-8820]),K([3414635,912600])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 56.1-b have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 8 & 4 & 2 & 4 \\
8 & 1 & 4 & 2 & 4 & 8 \\
8 & 4 & 1 & 2 & 4 & 8 \\
4 & 2 & 2 & 1 & 2 & 4 \\
2 & 4 & 4 & 2 & 1 & 2 \\
4 & 8 & 8 & 4 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 56.1-b over \(\Q(\sqrt{14}) \)
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sage:E.isogeny_class().curves
Isogeny class 56.1-b contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 56.1-b1
| \( \bigl[a\) , \( 1\) , \( a\) , \( -8820 a - 33003\) , \( 912600 a + 3414635\bigr] \)
|
| 56.1-b2
| \( \bigl[a\) , \( 1\) , \( a\) , \( -30 a + 112\) , \( -944 a + 3532\bigr] \)
|
| 56.1-b3
| \( \bigl[a\) , \( 1\) , \( a\) , \( 1770 a - 6623\) , \( 64686 a - 242033\bigr] \)
|
| 56.1-b4
| \( \bigl[a\) , \( 1\) , \( a\) , \( -570 a - 2133\) , \( 12630 a + 47257\bigr] \)
|
| 56.1-b5
| \( \bigl[a\) , \( 1\) , \( a\) , \( 8970 a - 33563\) , \( -881050 a + 3296587\bigr] \)
|
| 56.1-b6
| \( \bigl[a\) , \( 1\) , \( a\) , \( 8820 a - 33003\) , \( -912600 a + 3414635\bigr] \)
|
