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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 68413a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68413.a3 | 68413a1 | \([0, -1, 1, -6163, -182726]\) | \(4096000/37\) | \(233890432813\) | \([]\) | \(52416\) | \(1.0041\) | \(\Gamma_0(N)\)-optimal |
68413.a2 | 68413a2 | \([0, -1, 1, -43143, 3354411]\) | \(1404928000/50653\) | \(320196002520997\) | \([]\) | \(157248\) | \(1.5534\) | |
68413.a1 | 68413a3 | \([0, -1, 1, -3463793, 2482436292]\) | \(727057727488000/37\) | \(233890432813\) | \([]\) | \(471744\) | \(2.1027\) |
Rank
sage: E.rank()
The elliptic curves in class 68413a have rank \(0\).
Complex multiplication
The elliptic curves in class 68413a do not have complex multiplication.Modular form 68413.2.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 Cremona 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 Cremona labels.