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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 24986b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
24986.b3 | 24986b1 | \([1, 1, 0, 461, -6049]\) | \(12167/26\) | \(-23075095706\) | \([]\) | \(20160\) | \(0.67277\) | \(\Gamma_0(N)\)-optimal |
24986.b2 | 24986b2 | \([1, 1, 0, -4344, 217864]\) | \(-10218313/17576\) | \(-15598764697256\) | \([]\) | \(60480\) | \(1.2221\) | |
24986.b1 | 24986b3 | \([1, 1, 0, -441599, 112767301]\) | \(-10730978619193/6656\) | \(-5907224500736\) | \([]\) | \(181440\) | \(1.7714\) |
Rank
sage: E.rank()
The elliptic curves in class 24986b have rank \(2\).
Complex multiplication
The elliptic curves in class 24986b do not have complex multiplication.Modular form 24986.2.a.b
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.