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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 220409.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 220409.c1 | 220409c3 | \([1, -1, 0, -169328, 26855181]\) | \(209267191953/55223\) | \(141687109484207\) | \([2]\) | \(1036800\) | \(1.6998\) | |
| 220409.c2 | 220409c2 | \([1, -1, 0, -11893, 311640]\) | \(72511713/25921\) | \(66506194247689\) | \([2, 2]\) | \(518400\) | \(1.3533\) | |
| 220409.c3 | 220409c1 | \([1, -1, 0, -5048, -133285]\) | \(5545233/161\) | \(413081951849\) | \([2]\) | \(259200\) | \(1.0067\) | \(\Gamma_0(N)\)-optimal |
| 220409.c4 | 220409c4 | \([1, -1, 0, 36022, 2161159]\) | \(2014698447/1958887\) | \(-5025968108146783\) | \([2]\) | \(1036800\) | \(1.6998\) |
Rank
sage: E.rank()
The elliptic curves in class 220409.c have rank \(1\).
Complex multiplication
The elliptic curves in class 220409.c do not have complex multiplication.Modular form 220409.2.a.c
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.
