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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 331200bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331200.bc2 | 331200bc1 | \([0, 0, 0, 2397300, 719606000]\) | \(510273943271/370215360\) | \(-1105457141514240000000\) | \([2]\) | \(12386304\) | \(2.7262\) | \(\Gamma_0(N)\)-optimal |
331200.bc1 | 331200bc2 | \([0, 0, 0, -10850700, 6098294000]\) | \(47316161414809/22001657400\) | \(65696596969881600000000\) | \([2]\) | \(24772608\) | \(3.0728\) |
Rank
sage: E.rank()
The elliptic curves in class 331200bc have rank \(1\).
Complex multiplication
The elliptic curves in class 331200bc do not have complex multiplication.Modular form 331200.2.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.