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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 150075.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 150075.e1 | 150075e3 | \([1, -1, 1, -300150005, 2001575271872]\) | \(262537424941059264096001/250125\) | \(2849080078125\) | \([2]\) | \(9437184\) | \(3.0660\) | |
| 150075.e2 | 150075e2 | \([1, -1, 1, -18759380, 31278115622]\) | \(64096096056024006001/62562515625\) | \(712626154541015625\) | \([2, 2]\) | \(4718592\) | \(2.7195\) | |
| 150075.e3 | 150075e4 | \([1, -1, 1, -18618755, 31770021872]\) | \(-62665433378363916001/2004003001000125\) | \(-22826846683267048828125\) | \([2]\) | \(9437184\) | \(3.0660\) | |
| 150075.e4 | 150075e1 | \([1, -1, 1, -1181255, 481240622]\) | \(16003198512756001/488525390625\) | \(5564609527587890625\) | \([2]\) | \(2359296\) | \(2.3729\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 150075.e have rank \(0\).
Complex multiplication
The elliptic curves in class 150075.e do not have complex multiplication.Modular form 150075.2.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.
