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sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 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
\( x^{2} - 3 \); class number \(1\).
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sage:E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-41,10]),K([177,11])])
E.isogeny_class()
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sage:E.rank()
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magma:Rank(E);
The elliptic curves in class 768.1-c have
rank \( 1 \).
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sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 8 & 2 & 4 \\
8 & 1 & 2 & 4 & 4 & 8 \\
4 & 2 & 1 & 2 & 2 & 4 \\
8 & 4 & 2 & 1 & 4 & 8 \\
2 & 4 & 2 & 4 & 1 & 2 \\
4 & 8 & 4 & 8 & 2 & 1
\end{array}\right)\)
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sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 768.1-c over \(\Q(\sqrt{3}) \)
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sage:E.isogeny_class().curves
Isogeny class 768.1-c contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 768.1-c1
| \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 10 a - 41\) , \( 11 a + 177\bigr] \)
|
| 768.1-c2
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 200974 a - 348097\) , \( 64360541 a - 111475727\bigr] \)
|
| 768.1-c3
| \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 12804 a - 22177\) , \( 955655 a - 1655243\bigr] \)
|
| 768.1-c4
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -43196 a + 74818\) , \( 28011318 a - 48517026\bigr] \)
|
| 768.1-c5
| \( \bigl[0\) , \( -1\) , \( 0\) , \( 216 a - 374\) , \( -1926 a + 3336\bigr] \)
|
| 768.1-c6
| \( \bigl[0\) , \( -1\) , \( 0\) , \( -444 a + 766\) , \( -10806 a + 18720\bigr] \)
|
