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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 19110.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19110.p1 | 19110l3 | \([1, 1, 0, -2522202, 1540712916]\) | \(15082569606665230489/7751016000\) | \(911899281384000\) | \([2]\) | \(497664\) | \(2.2026\) | |
| 19110.p2 | 19110l4 | \([1, 1, 0, -2508482, 1558321164]\) | \(-14837772556740428569/342100087875000\) | \(-40247733238405875000\) | \([2]\) | \(995328\) | \(2.5492\) | |
| 19110.p3 | 19110l1 | \([1, 1, 0, -37167, 1222929]\) | \(48264326765929/22299191460\) | \(2623477576077540\) | \([2]\) | \(165888\) | \(1.6533\) | \(\Gamma_0(N)\)-optimal |
| 19110.p4 | 19110l2 | \([1, 1, 0, 130903, 9391131]\) | \(2108526614950391/1540302022350\) | \(-181214992627455150\) | \([2]\) | \(331776\) | \(1.9999\) |
Rank
sage: E.rank()
The elliptic curves in class 19110.p have rank \(1\).
Complex multiplication
The elliptic curves in class 19110.p do not have complex multiplication.Modular form 19110.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
