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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 1584m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1584.c2 | 1584m1 | \([0, 0, 0, -696, 8215]\) | \(-3196715008/649539\) | \(-7576222896\) | \([2]\) | \(960\) | \(0.61810\) | \(\Gamma_0(N)\)-optimal |
1584.c1 | 1584m2 | \([0, 0, 0, -11631, 482794]\) | \(932410994128/29403\) | \(5487305472\) | \([2]\) | \(1920\) | \(0.96467\) |
Rank
sage: E.rank()
The elliptic curves in class 1584m have rank \(1\).
Complex multiplication
The elliptic curves in class 1584m do not have complex multiplication.Modular form 1584.2.a.m
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.