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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 67081a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67081.g3 | 67081a1 | \([0, -1, 1, -223603, -40303880]\) | \(4096000/37\) | \(11168640412820317\) | \([]\) | \(344736\) | \(1.9019\) | \(\Gamma_0(N)\)-optimal |
67081.g2 | 67081a2 | \([0, -1, 1, -1565223, 730389729]\) | \(1404928000/50653\) | \(15289868725151013973\) | \([]\) | \(1034208\) | \(2.4512\) | |
67081.g1 | 67081a3 | \([0, -1, 1, -125665073, 542254977186]\) | \(727057727488000/37\) | \(11168640412820317\) | \([]\) | \(3102624\) | \(3.0005\) |
Rank
sage: E.rank()
The elliptic curves in class 67081a have rank \(0\).
Complex multiplication
The elliptic curves in class 67081a do not have complex multiplication.Modular form 67081.2.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.