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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 1638.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1638.s1 | 1638q3 | \([1, -1, 1, -17609, -894787]\) | \(828279937799497/193444524\) | \(141021057996\) | \([2]\) | \(3072\) | \(1.1299\) | |
| 1638.s2 | 1638q2 | \([1, -1, 1, -1229, -10267]\) | \(281397674377/96589584\) | \(70413806736\) | \([2, 2]\) | \(1536\) | \(0.78338\) | |
| 1638.s3 | 1638q1 | \([1, -1, 1, -509, 4421]\) | \(19968681097/628992\) | \(458535168\) | \([4]\) | \(768\) | \(0.43680\) | \(\Gamma_0(N)\)-optimal |
| 1638.s4 | 1638q4 | \([1, -1, 1, 3631, -74419]\) | \(7264187703863/7406095788\) | \(-5399043829452\) | \([2]\) | \(3072\) | \(1.1299\) |
Rank
sage: E.rank()
The elliptic curves in class 1638.s have rank \(0\).
Complex multiplication
The elliptic curves in class 1638.s do not have complex multiplication.Modular form 1638.2.a.s
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.
