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SageMath
E = EllipticCurve("qm1")
E.isogeny_class()
Elliptic curves in class 486720qm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.qm2 | 486720qm1 | \([0, 0, 0, 448188, 190102016]\) | \(314432/675\) | \(-21373784182656921600\) | \([2]\) | \(13418496\) | \(2.3933\) | \(\Gamma_0(N)\)-optimal* |
486720.qm1 | 486720qm2 | \([0, 0, 0, -3506412, 2061418736]\) | \(18821096/3645\) | \(923347476690779013120\) | \([2]\) | \(26836992\) | \(2.7399\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720qm have rank \(1\).
Complex multiplication
The elliptic curves in class 486720qm do not have complex multiplication.Modular form 486720.2.a.qm
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.