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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 10192.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10192.d1 | 10192z2 | \([0, 1, 0, -268, -1800]\) | \(-170338000/2197\) | \(-27559168\) | \([]\) | \(2592\) | \(0.23671\) | |
| 10192.d2 | 10192z1 | \([0, 1, 0, 12, -8]\) | \(14000/13\) | \(-163072\) | \([]\) | \(864\) | \(-0.31260\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10192.d have rank \(1\).
Complex multiplication
The elliptic curves in class 10192.d do not have complex multiplication.Modular form 10192.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
