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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 236992.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
236992.bd1 | 236992bd2 | \([0, 0, 0, -8064076, -8786520720]\) | \(1494447319737/5411854\) | \(210016303324386033664\) | \([2]\) | \(9732096\) | \(2.7605\) | |
236992.bd2 | 236992bd1 | \([0, 0, 0, -277196, -261444496]\) | \(-60698457/725788\) | \(-28165451757789380608\) | \([2]\) | \(4866048\) | \(2.4139\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 236992.bd have rank \(2\).
Complex multiplication
The elliptic curves in class 236992.bd do not have complex multiplication.Modular form 236992.2.a.bd
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.