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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 75150bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75150.bb2 | 75150bc1 | \([1, -1, 1, -2105, 23897]\) | \(2444008923/855040\) | \(360720000000\) | \([2]\) | \(76800\) | \(0.91904\) | \(\Gamma_0(N)\)-optimal |
75150.bb1 | 75150bc2 | \([1, -1, 1, -14105, -624103]\) | \(735580702683/22311200\) | \(9412537500000\) | \([2]\) | \(153600\) | \(1.2656\) |
Rank
sage: E.rank()
The elliptic curves in class 75150bc have rank \(1\).
Complex multiplication
The elliptic curves in class 75150bc do not have complex multiplication.Modular form 75150.2.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.