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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 19110by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19110.cd3 | 19110by1 | \([1, 1, 1, -295, -12643]\) | \(-24137569/561600\) | \(-66071678400\) | \([2]\) | \(17280\) | \(0.75690\) | \(\Gamma_0(N)\)-optimal |
| 19110.cd2 | 19110by2 | \([1, 1, 1, -10095, -392883]\) | \(967068262369/4928040\) | \(579778977960\) | \([2]\) | \(34560\) | \(1.1035\) | |
| 19110.cd4 | 19110by3 | \([1, 1, 1, 2645, 331925]\) | \(17394111071/411937500\) | \(-48464034937500\) | \([2]\) | \(51840\) | \(1.3062\) | |
| 19110.cd1 | 19110by4 | \([1, 1, 1, -58605, 5158425]\) | \(189208196468929/10860320250\) | \(1277705817092250\) | \([2]\) | \(103680\) | \(1.6528\) |
Rank
sage: E.rank()
The elliptic curves in class 19110by have rank \(0\).
Complex multiplication
The elliptic curves in class 19110by do not have complex multiplication.Modular form 19110.2.a.by
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 Cremona 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 Cremona labels.
