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SageMath
E = EllipticCurve("gc1")
E.isogeny_class()
Elliptic curves in class 473200.gc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
473200.gc1 | 473200gc2 | \([0, 1, 0, -4192608, 3352668788]\) | \(-156116857/2744\) | \(-143255366299136000000\) | \([]\) | \(14556672\) | \(2.6649\) | \(\Gamma_0(N)\)-optimal* |
473200.gc2 | 473200gc1 | \([0, 1, 0, 201392, 22016788]\) | \(17303/14\) | \(-730894726016000000\) | \([]\) | \(4852224\) | \(2.1156\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 473200.gc have rank \(1\).
Complex multiplication
The elliptic curves in class 473200.gc do not have complex multiplication.Modular form 473200.2.a.gc
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 LMFDB 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 LMFDB labels.