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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 38640bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38640.bd1 | 38640bt1 | \([0, -1, 0, -3680, 75072]\) | \(1345938541921/203765625\) | \(834624000000\) | \([2]\) | \(49152\) | \(1.0115\) | \(\Gamma_0(N)\)-optimal |
38640.bd2 | 38640bt2 | \([0, -1, 0, 6320, 403072]\) | \(6814692748079/21258460125\) | \(-87074652672000\) | \([2]\) | \(98304\) | \(1.3581\) |
Rank
sage: E.rank()
The elliptic curves in class 38640bt have rank \(1\).
Complex multiplication
The elliptic curves in class 38640bt do not have complex multiplication.Modular form 38640.2.a.bt
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.