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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 17640e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
17640.o2 | 17640e1 | \([0, 0, 0, -792918, 307613033]\) | \(-1084767227025408/176547030625\) | \(-8972891253792270000\) | \([2]\) | \(368640\) | \(2.3643\) | \(\Gamma_0(N)\)-optimal |
17640.o1 | 17640e2 | \([0, 0, 0, -13146063, 18345675362]\) | \(308971819397054448/6565234375\) | \(5338782206100000000\) | \([2]\) | \(737280\) | \(2.7109\) |
Rank
sage: E.rank()
The elliptic curves in class 17640e have rank \(0\).
Complex multiplication
The elliptic curves in class 17640e do not have complex multiplication.Modular form 17640.2.a.e
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.