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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([0,1]),K([0,1]),K([0,0]),K([1740,6444]),K([-200320,-79392])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 8450.9-a 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 8450.9-a over \(\Q(\sqrt{-1}) \)
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sage:E.isogeny_class().curves
Isogeny class 8450.9-a contains
6 curves linked by isogenies of
degrees dividing 18.
| Curve label |
Weierstrass Coefficients |
| 8450.9-a1
| \( \bigl[i\) , \( i\) , \( 0\) , \( 6444 i + 1740\) , \( -79392 i - 200320\bigr] \)
|
| 8450.9-a2
| \( \bigl[i\) , \( i\) , \( 0\) , \( 524 i - 445\) , \( 6620 i - 2129\bigr] \)
|
| 8450.9-a3
| \( \bigl[i\) , \( i\) , \( 0\) , \( -266 i - 915\) , \( 19254 i + 12233\bigr] \)
|
| 8450.9-a4
| \( \bigl[i\) , \( i\) , \( 0\) , \( 2684 i + 8060\) , \( -553840 i - 196784\bigr] \)
|
| 8450.9-a5
| \( \bigl[i\) , \( i\) , \( 0\) , \( -26 i + 30\) , \( 54 i + 8\bigr] \)
|
| 8450.9-a6
| \( \bigl[i\) , \( i\) , \( 0\) , \( -261 i + 425\) , \( 3077 i + 3197\bigr] \)
|
