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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 25410b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25410.f2 | 25410b1 | \([1, 1, 0, -178, 928]\) | \(-472729139/52500\) | \(-69877500\) | \([2]\) | \(11520\) | \(0.24261\) | \(\Gamma_0(N)\)-optimal |
25410.f1 | 25410b2 | \([1, 1, 0, -2928, 59778]\) | \(2086847005139/22050\) | \(29348550\) | \([2]\) | \(23040\) | \(0.58918\) |
Rank
sage: E.rank()
The elliptic curves in class 25410b have rank \(1\).
Complex multiplication
The elliptic curves in class 25410b do not have complex multiplication.Modular form 25410.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.