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SageMath
E = EllipticCurve("fp1")
E.isogeny_class()
Elliptic curves in class 25200fp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25200.du1 | 25200fp1 | \([0, 0, 0, -24555, 1473050]\) | \(4386781853/27216\) | \(10158317568000\) | \([2]\) | \(61440\) | \(1.3338\) | \(\Gamma_0(N)\)-optimal |
25200.du2 | 25200fp2 | \([0, 0, 0, -10155, 3186650]\) | \(-310288733/11573604\) | \(-4319824545792000\) | \([2]\) | \(122880\) | \(1.6804\) |
Rank
sage: E.rank()
The elliptic curves in class 25200fp have rank \(1\).
Complex multiplication
The elliptic curves in class 25200fp do not have complex multiplication.Modular form 25200.2.a.fp
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.