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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 382200ej
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
382200.ej2 | 382200ej1 | \([0, -1, 0, -67538263508, -6670766104172988]\) | \(211072197308055014773168/3052652281946850375\) | \(492742121973345579682210500000000\) | \([2]\) | \(1759887360\) | \(5.0770\) | \(\Gamma_0(N)\)-optimal |
382200.ej1 | 382200ej2 | \([0, -1, 0, -131215013008, 7911846298822012]\) | \(386965237776463086681532/182055746334444328125\) | \(117545696634749711688564750000000000\) | \([2]\) | \(3519774720\) | \(5.4236\) |
Rank
sage: E.rank()
The elliptic curves in class 382200ej have rank \(1\).
Complex multiplication
The elliptic curves in class 382200ej do not have complex multiplication.Modular form 382200.2.a.ej
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.