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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 121.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121.d1 | 121d3 | \([0, -1, 1, -946260, 354609639]\) | \(-52893159101157376/11\) | \(-19487171\) | \([]\) | \(600\) | \(1.6957\) | |
| 121.d2 | 121d2 | \([0, -1, 1, -1250, 31239]\) | \(-122023936/161051\) | \(-285311670611\) | \([]\) | \(120\) | \(0.89094\) | |
| 121.d3 | 121d1 | \([0, -1, 1, -40, -221]\) | \(-4096/11\) | \(-19487171\) | \([]\) | \(24\) | \(0.086219\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121.d have rank \(0\).
Complex multiplication
The elliptic curves in class 121.d do not have complex multiplication.Modular form 121.2.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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
