Show commands:
SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 323400.em
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 323400.em1 | 323400em2 | \([0, -1, 0, -81608, -8944788]\) | \(5476248398/891\) | \(9779616000000\) | \([2]\) | \(1310720\) | \(1.5016\) | |
| 323400.em2 | 323400em1 | \([0, -1, 0, -4608, -166788]\) | \(-1972156/1089\) | \(-5976432000000\) | \([2]\) | \(655360\) | \(1.1551\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 323400.em have rank \(1\).
Complex multiplication
The elliptic curves in class 323400.em do not have complex multiplication.Modular form 323400.2.a.em
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
