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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1650b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1650.c3 | 1650b1 | \([1, 1, 0, -1125, 10125]\) | \(10091699281/2737152\) | \(42768000000\) | \([2]\) | \(1600\) | \(0.74785\) | \(\Gamma_0(N)\)-optimal |
| 1650.c4 | 1650b2 | \([1, 1, 0, 2875, 70125]\) | \(168105213359/228637728\) | \(-3572464500000\) | \([2]\) | \(3200\) | \(1.0944\) | |
| 1650.c1 | 1650b3 | \([1, 1, 0, -251625, -48687375]\) | \(112763292123580561/1932612\) | \(30197062500\) | \([2]\) | \(8000\) | \(1.5526\) | |
| 1650.c2 | 1650b4 | \([1, 1, 0, -251375, -48788625]\) | \(-112427521449300721/466873642818\) | \(-7294900669031250\) | \([2]\) | \(16000\) | \(1.8991\) |
Rank
sage: E.rank()
The elliptic curves in class 1650b have rank \(0\).
Complex multiplication
The elliptic curves in class 1650b do not have complex multiplication.Modular form 1650.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
