Show commands:
SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 106722hl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.eh2 | 106722hl1 | \([1, -1, 1, 158971, -9298803]\) | \(24167/16\) | \(-294155675604329616\) | \([]\) | \(1520640\) | \(2.0407\) | \(\Gamma_0(N)\)-optimal |
| 106722.eh1 | 106722hl2 | \([1, -1, 1, -2775884, -1827734961]\) | \(-128667913/4096\) | \(-75303852954708381696\) | \([]\) | \(4561920\) | \(2.5900\) |
Rank
sage: E.rank()
The elliptic curves in class 106722hl have rank \(1\).
Complex multiplication
The elliptic curves in class 106722hl do not have complex multiplication.Modular form 106722.2.a.hl
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
