Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 1584.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1584.p1 | 1584o2 | \([0, 0, 0, -696, -7108]\) | \(-199794688/1331\) | \(-248396544\) | \([]\) | \(720\) | \(0.44632\) | |
| 1584.p2 | 1584o1 | \([0, 0, 0, 24, -52]\) | \(8192/11\) | \(-2052864\) | \([]\) | \(240\) | \(-0.10299\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1584.p have rank \(1\).
Complex multiplication
The elliptic curves in class 1584.p do not have complex multiplication.Modular form 1584.2.a.p
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.
