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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -3, -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
\( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).
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sage:E = EllipticCurve([K([-1,0,1]),K([-1,-2,1]),K([-1,0,1]),K([23609,-59948,-51231]),K([-2734899,6944424,5934968])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 400.2-e have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrrrrrr}
1 & 3 & 6 & 2 & 2 & 6 & 4 & 12 & 2 & 6 & 4 & 12 \\
3 & 1 & 2 & 6 & 6 & 2 & 12 & 4 & 6 & 2 & 12 & 4 \\
6 & 2 & 1 & 3 & 12 & 4 & 24 & 8 & 12 & 4 & 24 & 8 \\
2 & 6 & 3 & 1 & 4 & 12 & 8 & 24 & 4 & 12 & 8 & 24 \\
2 & 6 & 12 & 4 & 1 & 3 & 8 & 24 & 4 & 12 & 8 & 24 \\
6 & 2 & 4 & 12 & 3 & 1 & 24 & 8 & 12 & 4 & 24 & 8 \\
4 & 12 & 24 & 8 & 8 & 24 & 1 & 3 & 2 & 6 & 4 & 12 \\
12 & 4 & 8 & 24 & 24 & 8 & 3 & 1 & 6 & 2 & 12 & 4 \\
2 & 6 & 12 & 4 & 4 & 12 & 2 & 6 & 1 & 3 & 2 & 6 \\
6 & 2 & 4 & 12 & 12 & 4 & 6 & 2 & 3 & 1 & 6 & 2 \\
4 & 12 & 24 & 8 & 8 & 24 & 4 & 12 & 2 & 6 & 1 & 3 \\
12 & 4 & 8 & 24 & 24 & 8 & 12 & 4 & 6 & 2 & 3 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 400.2-e over 3.3.148.1
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sage:E.isogeny_class().curves
Isogeny class 400.2-e contains
12 curves linked by isogenies of
degrees dividing 24.
| Curve label |
Weierstrass Coefficients |
| 400.2-e1
| \( \bigl[a^{2} - 1\) , \( a^{2} - 2 a - 1\) , \( a^{2} - 1\) , \( -51231 a^{2} - 59948 a + 23609\) , \( 5934968 a^{2} + 6944424 a - 2734899\bigr] \)
|
| 400.2-e2
| \( \bigl[a^{2} - 1\) , \( a^{2} - 2 a - 1\) , \( a^{2} - 1\) , \( -3561151 a^{2} - 4166858 a + 1641019\) , \( 6735634702 a^{2} + 7881275142 a - 3103855419\bigr] \)
|
| 400.2-e3
| \( \bigl[a^{2} - 1\) , \( -a + 1\) , \( a^{2} - a - 2\) , \( -39 a^{2} + 41 a - 11\) , \( -90 a^{2} + 410 a - 117\bigr] \)
|
| 400.2-e4
| \( \bigl[a^{2} - 1\) , \( -a + 1\) , \( a^{2} - a - 2\) , \( a^{2} + a - 1\) , \( a^{2} + 2 a - 2\bigr] \)
|
| 400.2-e5
| \( \bigl[a + 1\) , \( -a - 1\) , \( a^{2} - a - 2\) , \( -8323587 a^{2} - 9739316 a + 3835603\) , \( 24073033070 a^{2} + 28167530687 a - 11093121505\bigr] \)
|
| 400.2-e6
| \( \bigl[a + 1\) , \( -a - 1\) , \( a^{2} - a - 2\) , \( -674134712 a^{2} - 788795916 a + 310648778\) , \( 17552202654819 a^{2} + 20537595136925 a - 8088250290020\bigr] \)
|
| 400.2-e7
| \( \bigl[a + 1\) , \( -a - 1\) , \( a^{2} - 1\) , \( 105815 a^{2} + 123813 a - 48760\) , \( -28888097400 a^{2} - 33801572392 a + 13311956725\bigr] \)
|
| 400.2-e8
| \( \bigl[a + 1\) , \( -a - 1\) , \( a^{2} - 1\) , \( -952345 a^{2} - 1114327 a + 438850\) , \( 779979186568 a^{2} + 912643106054 a - 359423088145\bigr] \)
|
| 400.2-e9
| \( \bigl[a + 1\) , \( -a - 1\) , \( a^{2} - 1\) , \( -4622250 a^{2} - 5408432 a + 2129985\) , \( -9795739727 a^{2} - 11461862681 a + 4513985865\bigr] \)
|
| 400.2-e10
| \( \bigl[a + 1\) , \( -a - 1\) , \( a^{2} - 1\) , \( -42314515 a^{2} - 49511642 a + 19499000\) , \( 271783667397 a^{2} + 318010396507 a - 125240938125\bigr] \)
|
| 400.2-e11
| \( \bigl[a + 1\) , \( -a - 1\) , \( a^{2} - 1\) , \( -73608095 a^{2} - 86127837 a + 33919430\) , \( -633264406060 a^{2} - 740974123992 a + 291815284765\bigr] \)
|
| 400.2-e12
| \( \bigl[a + 1\) , \( -a - 1\) , \( a^{2} - 1\) , \( -86574145 a^{2} - 101299237 a + 39894330\) , \( -394902257472 a^{2} - 462069795006 a + 181975354395\bigr] \)
|
