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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 485520e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.e1 | 485520e1 | \([0, -1, 0, -7456296, -48450168720]\) | \(-1604507946409/34566497280\) | \(-987658242499650602926080\) | \([]\) | \(63452160\) | \(3.2850\) | \(\Gamma_0(N)\)-optimal* |
485520.e2 | 485520e2 | \([0, -1, 0, 66828264, 1278093789936]\) | \(1155188682445031/25367150592000\) | \(-724808046588360929574912000\) | \([]\) | \(190356480\) | \(3.8343\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485520e have rank \(1\).
Complex multiplication
The elliptic curves in class 485520e do not have complex multiplication.Modular form 485520.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.