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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 20160fc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20160.ep2 | 20160fc1 | \([0, 0, 0, 933, 19024]\) | \(1925134784/4465125\) | \(-208324872000\) | \([2]\) | \(18432\) | \(0.85590\) | \(\Gamma_0(N)\)-optimal |
20160.ep1 | 20160fc2 | \([0, 0, 0, -7572, 209536]\) | \(16079333824/2953125\) | \(8817984000000\) | \([2]\) | \(36864\) | \(1.2025\) |
Rank
sage: E.rank()
The elliptic curves in class 20160fc have rank \(1\).
Complex multiplication
The elliptic curves in class 20160fc do not have complex multiplication.Modular form 20160.2.a.fc
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.