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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 37440.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37440.cs1 | 37440bv4 | \([0, 0, 0, -49840428, -135431833552]\) | \(71647584155243142409/10140000\) | \(1937784176640000\) | \([2]\) | \(1966080\) | \(2.7863\) | |
| 37440.cs2 | 37440bv3 | \([0, 0, 0, -3576108, -1448667088]\) | \(26465989780414729/10571870144160\) | \(2020315846434525020160\) | \([2]\) | \(1966080\) | \(2.7863\) | |
| 37440.cs3 | 37440bv2 | \([0, 0, 0, -3115308, -2115721168]\) | \(17496824387403529/6580454400\) | \(1257544419272294400\) | \([2, 2]\) | \(983040\) | \(2.4397\) | |
| 37440.cs4 | 37440bv1 | \([0, 0, 0, -166188, -43079632]\) | \(-2656166199049/2658140160\) | \(-507978495201116160\) | \([2]\) | \(491520\) | \(2.0931\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 37440.cs have rank \(1\).
Complex multiplication
The elliptic curves in class 37440.cs do not have complex multiplication.Modular form 37440.2.a.cs
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
