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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 397800bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
397800.bc1 | 397800bc1 | \([0, 0, 0, -20550, -963875]\) | \(5266130944/845325\) | \(154060481250000\) | \([2]\) | \(786432\) | \(1.4447\) | \(\Gamma_0(N)\)-optimal |
397800.bc2 | 397800bc2 | \([0, 0, 0, 36825, -5381750]\) | \(1893932336/5386875\) | \(-15708127500000000\) | \([2]\) | \(1572864\) | \(1.7912\) |
Rank
sage: E.rank()
The elliptic curves in class 397800bc have rank \(1\).
Complex multiplication
The elliptic curves in class 397800bc do not have complex multiplication.Modular form 397800.2.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.