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SageMath
E = EllipticCurve("gc1")
E.isogeny_class()
Elliptic curves in class 97344gc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
97344.gk2 | 97344gc1 | \([0, 0, 0, -32448, 2021240]\) | \(1048576/117\) | \(421573652517888\) | \([2]\) | \(516096\) | \(1.5393\) | \(\Gamma_0(N)\)-optimal |
97344.gk1 | 97344gc2 | \([0, 0, 0, -123708, -14588080]\) | \(3631696/507\) | \(29229106574573568\) | \([2]\) | \(1032192\) | \(1.8858\) |
Rank
sage: E.rank()
The elliptic curves in class 97344gc have rank \(1\).
Complex multiplication
The elliptic curves in class 97344gc do not have complex multiplication.Modular form 97344.2.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 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.