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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 101640.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 101640.e1 | 101640br4 | \([0, -1, 0, -1971658256, 29845781684556]\) | \(233632133015204766393938/29145526885986328125\) | \(105744545291601562500000000000\) | \([2]\) | \(117964800\) | \(4.2989\) | |
| 101640.e2 | 101640br2 | \([0, -1, 0, -492491336, -3723615731460]\) | \(7282213870869695463556/912102595400390625\) | \(1654625675274354090000000000\) | \([2, 2]\) | \(58982400\) | \(3.9523\) | |
| 101640.e3 | 101640br1 | \([0, -1, 0, -476613716, -4004732169084]\) | \(26401417552259125806544/507547744790625\) | \(230182858319110250400000\) | \([2]\) | \(29491200\) | \(3.6057\) | \(\Gamma_0(N)\)-optimal |
| 101640.e4 | 101640br3 | \([0, -1, 0, 732633664, -19301815181460]\) | \(11986661998777424518222/51295853620928503125\) | \(-186109406692445634405638400000\) | \([2]\) | \(117964800\) | \(4.2989\) |
Rank
sage: E.rank()
The elliptic curves in class 101640.e have rank \(1\).
Complex multiplication
The elliptic curves in class 101640.e do not have complex multiplication.Modular form 101640.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
