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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 592.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
592.a1 | 592e3 | \([0, -1, 0, -29973, 2007325]\) | \(727057727488000/37\) | \(151552\) | \([]\) | \(432\) | \(0.91523\) | |
592.a2 | 592e2 | \([0, -1, 0, -373, 2813]\) | \(1404928000/50653\) | \(207474688\) | \([]\) | \(144\) | \(0.36592\) | |
592.a3 | 592e1 | \([0, -1, 0, -53, -131]\) | \(4096000/37\) | \(151552\) | \([]\) | \(48\) | \(-0.18338\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 592.a have rank \(1\).
Complex multiplication
The elliptic curves in class 592.a do not have complex multiplication.Modular form 592.2.a.a
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 LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.