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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 277725bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277725.bc1 | 277725bc1 | \([1, 0, 0, -9533, -347568]\) | \(5177717/189\) | \(3497347877625\) | \([2]\) | \(608256\) | \(1.1765\) | \(\Gamma_0(N)\)-optimal |
277725.bc2 | 277725bc2 | \([1, 0, 0, 3692, -1233643]\) | \(300763/35721\) | \(-660998748871125\) | \([2]\) | \(1216512\) | \(1.5231\) |
Rank
sage: E.rank()
The elliptic curves in class 277725bc have rank \(1\).
Complex multiplication
The elliptic curves in class 277725bc do not have complex multiplication.Modular form 277725.2.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 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.