![Copy content]()
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))
![Copy content]()
pari:K = nfinit(Polrev(%s));
![Copy content]()
magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R!%s);
![Copy content]()
oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx(%s))
Generator \(a\), with minimal polynomial
\( x^{2} - 6 \); class number \(1\).
![Copy content]()
sage:E = EllipticCurve([K([0,1]),K([-1,1]),K([0,0]),K([194,-78]),K([-10718,4376])])
E.isogeny_class()
![Copy content]()
sage:E.rank()
![Copy content]()
magma:Rank(E);
The elliptic curves in class 24.1-b have
rank \( 1 \).
![Copy content]()
sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 2 & 8 & 4 \\
8 & 1 & 2 & 4 & 4 & 8 \\
4 & 2 & 1 & 2 & 2 & 4 \\
2 & 4 & 2 & 1 & 4 & 2 \\
8 & 4 & 2 & 4 & 1 & 8 \\
4 & 8 & 4 & 2 & 8 & 1
\end{array}\right)\)
![Copy content]()
sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 24.1-b over \(\Q(\sqrt{6}) \)
![Copy content]()
sage:E.isogeny_class().curves
Isogeny class 24.1-b contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 24.1-b1
| \( \bigl[a\) , \( a - 1\) , \( 0\) , \( -78 a + 194\) , \( 4376 a - 10718\bigr] \)
|
| 24.1-b2
| \( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \)
|
| 24.1-b3
| \( \bigl[a\) , \( a - 1\) , \( 0\) , \( 22 a - 51\) , \( -78 a + 192\bigr] \)
|
| 24.1-b4
| \( \bigl[a\) , \( a - 1\) , \( 0\) , \( 122 a - 296\) , \( 1012 a - 2478\bigr] \)
|
| 24.1-b5
| \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -322 a - 786\) , \( 5124 a + 12552\bigr] \)
|
| 24.1-b6
| \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -1922 a - 4706\) , \( -70528 a - 172758\bigr] \)
|
