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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 7056.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7056.bc1 | 7056q2 | \([0, 0, 0, -56595, -4605118]\) | \(665500/81\) | \(2440028303096832\) | \([2]\) | \(28672\) | \(1.6830\) | |
7056.bc2 | 7056q1 | \([0, 0, 0, 5145, -369754]\) | \(2000/9\) | \(-67778563974912\) | \([2]\) | \(14336\) | \(1.3364\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7056.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 7056.bc do not have complex multiplication.Modular form 7056.2.a.bc
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.