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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -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
\( x^{2} - x + 2 \); class number \(1\).
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sage:E = EllipticCurve([K([1,0]),K([1,0]),K([0,0]),K([-850,0]),K([-27125,0])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 39375.1-d have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 8 & 4 & 16 & 8 & 16 & 2 & 4 \\
8 & 1 & 2 & 2 & 4 & 2 & 4 & 8 \\
4 & 2 & 1 & 4 & 2 & 4 & 2 & 4 \\
16 & 2 & 4 & 1 & 8 & 4 & 8 & 16 \\
8 & 4 & 2 & 8 & 1 & 8 & 4 & 8 \\
16 & 2 & 4 & 4 & 8 & 1 & 8 & 16 \\
2 & 4 & 2 & 8 & 4 & 8 & 1 & 2 \\
4 & 8 & 4 & 16 & 8 & 16 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 39375.1-d over \(\Q(\sqrt{-7}) \)
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sage:E.isogeny_class().curves
Isogeny class 39375.1-d contains
8 curves linked by isogenies of
degrees dividing 16.
| Curve label |
Weierstrass Coefficients |
| 39375.1-d1
| \( \bigl[1\) , \( 1\) , \( 0\) , \( -850\) , \( -27125\bigr] \)
|
| 39375.1-d2
| \( \bigl[1\) , \( 1\) , \( 0\) , \( 25\) , \( 0\bigr] \)
|
| 39375.1-d3
| \( \bigl[1\) , \( 1\) , \( 0\) , \( -100\) , \( -125\bigr] \)
|
| 39375.1-d4
| \( \bigl[1\) , \( -a + 1\) , \( a\) , \( -51 a + 41\) , \( 49 a - 246\bigr] \)
|
| 39375.1-d5
| \( \bigl[1\) , \( 1\) , \( 0\) , \( -975\) , \( 11250\bigr] \)
|
| 39375.1-d6
| \( \bigl[1\) , \( a\) , \( a + 1\) , \( 50 a - 10\) , \( -50 a - 197\bigr] \)
|
| 39375.1-d7
| \( \bigl[1\) , \( 1\) , \( 0\) , \( -1225\) , \( -17000\bigr] \)
|
| 39375.1-d8
| \( \bigl[1\) , \( 1\) , \( 0\) , \( -19600\) , \( -1064375\bigr] \)
|
