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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 34650cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34650.cu1 | 34650cm1 | \([1, -1, 1, -4708505, -3930858503]\) | \(37537160298467283/5519360000\) | \(1697461920000000000\) | \([2]\) | \(1032192\) | \(2.5122\) | \(\Gamma_0(N)\)-optimal |
34650.cu2 | 34650cm2 | \([1, -1, 1, -4276505, -4681674503]\) | \(-28124139978713043/14526050000000\) | \(-4467441283593750000000\) | \([2]\) | \(2064384\) | \(2.8588\) |
Rank
sage: E.rank()
The elliptic curves in class 34650cm have rank \(0\).
Complex multiplication
The elliptic curves in class 34650cm do not have complex multiplication.Modular form 34650.2.a.cm
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.