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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([161, 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
\( x^{2} + 161 \); class number \(16\).
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sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([343,0]),K([851,0])])
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 14.1-c 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 14.1-c contains
6 curves linked by isogenies of
degrees dividing 18.
| Curve label |
Weierstrass Coefficients |
| 14.1-c1
| \( \bigl[a\) , \( 1\) , \( 0\) , \( 343\) , \( 851\bigr] \)
|
| 14.1-c2
| \( \bigl[a\) , \( 1\) , \( 0\) , \( 513\) , \( -2233\bigr] \)
|
| 14.1-c3
| \( \bigl[a\) , \( 1\) , \( 0\) , \( 518\) , \( -2292\bigr] \)
|
| 14.1-c4
| \( \bigl[a\) , \( 1\) , \( 0\) , \( 478\) , \( -1708\bigr] \)
|
| 14.1-c5
| \( \bigl[a\) , \( 1\) , \( 0\) , \( 503\) , \( -2115\bigr] \)
|
| 14.1-c6
| \( \bigl[a\) , \( 1\) , \( 0\) , \( -2217\) , \( 88403\bigr] \)
|
