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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 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 \(i\), with minimal polynomial
\( x^{2} + 1 \); class number \(1\).
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sage:E = EllipticCurve([K([1,1]),K([-1,0]),K([1,1]),K([-2036,267]),K([36407,-7245])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 5200.1-b have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 3 & 6 & 2 & 9 & 18 \\
3 & 1 & 2 & 6 & 3 & 6 \\
6 & 2 & 1 & 3 & 6 & 3 \\
2 & 6 & 3 & 1 & 18 & 9 \\
9 & 3 & 6 & 18 & 1 & 2 \\
18 & 6 & 3 & 9 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 5200.1-b over \(\Q(\sqrt{-1}) \)
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sage:E.isogeny_class().curves
Isogeny class 5200.1-b contains
6 curves linked by isogenies of
degrees dividing 18.
| Curve label |
Weierstrass Coefficients |
| 5200.1-b1
| \( \bigl[i + 1\) , \( -1\) , \( i + 1\) , \( 267 i - 2036\) , \( -7245 i + 36407\bigr] \)
|
| 5200.1-b2
| \( \bigl[i + 1\) , \( -1\) , \( i + 1\) , \( 187 i - 96\) , \( 1127 i + 111\bigr] \)
|
| 5200.1-b3
| \( \bigl[i + 1\) , \( -1\) , \( i + 1\) , \( 227 i + 184\) , \( 2831 i - 2761\bigr] \)
|
| 5200.1-b4
| \( \bigl[i + 1\) , \( -1\) , \( i + 1\) , \( -1973 i - 1716\) , \( -86413 i + 51831\bigr] \)
|
| 5200.1-b5
| \( \bigl[i + 1\) , \( -1\) , \( i + 1\) , \( -13 i + 4\) , \( 11 i - 1\bigr] \)
|
| 5200.1-b6
| \( \bigl[i + 1\) , \( -1\) , \( i + 1\) , \( -153 i + 24\) , \( 439 i - 605\bigr] \)
|
