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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 464968.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
464968.p1 | 464968p2 | \([0, -1, 0, -6125568, -848749012]\) | \(263822189935250/149429406721\) | \(14397518001145377179648\) | \([2]\) | \(25159680\) | \(2.9417\) | \(\Gamma_0(N)\)-optimal* |
464968.p2 | 464968p1 | \([0, -1, 0, 1513192, -106261540]\) | \(7953970437500/4703287687\) | \(-226580800339320060928\) | \([2]\) | \(12579840\) | \(2.5952\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 464968.p have rank \(1\).
Complex multiplication
The elliptic curves in class 464968.p do not have complex multiplication.Modular form 464968.2.a.p
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 LMFDB 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 LMFDB labels.