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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 6050bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6050.bc1 | 6050bc1 | \([1, 1, 1, -3088, -181719]\) | \(-117649/440\) | \(-12179481875000\) | \([]\) | \(11520\) | \(1.1966\) | \(\Gamma_0(N)\)-optimal |
6050.bc2 | 6050bc2 | \([1, 1, 1, 27162, 4295281]\) | \(80062991/332750\) | \(-9210733167968750\) | \([]\) | \(34560\) | \(1.7459\) |
Rank
sage: E.rank()
The elliptic curves in class 6050bc have rank \(1\).
Complex multiplication
The elliptic curves in class 6050bc do not have complex multiplication.Modular form 6050.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.