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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 83490.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 83490.e1 | 83490c4 | \([1, 1, 0, -5325333, -4732296213]\) | \(9427749584548611889/2308688250\) | \(4089982064858250\) | \([2]\) | \(1843200\) | \(2.3724\) | |
| 83490.e2 | 83490c2 | \([1, 1, 0, -334083, -73463463]\) | \(2327730853071889/36005062500\) | \(63785164527562500\) | \([2, 2]\) | \(921600\) | \(2.0258\) | |
| 83490.e3 | 83490c1 | \([1, 1, 0, -41263, 1439893]\) | \(4385977971409/2020458000\) | \(3579364594938000\) | \([2]\) | \(460800\) | \(1.6792\) | \(\Gamma_0(N)\)-optimal |
| 83490.e4 | 83490c3 | \([1, 1, 0, -27953, -202589097]\) | \(-1363569097969/10006347656250\) | \(-17726855260253906250\) | \([2]\) | \(1843200\) | \(2.3724\) |
Rank
sage: E.rank()
The elliptic curves in class 83490.e have rank \(2\).
Complex multiplication
The elliptic curves in class 83490.e do not have complex multiplication.Modular form 83490.2.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.
