Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 59584.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 59584.z1 | 59584cl3 | \([0, -1, 0, -268193, -429928127]\) | \(-69173457625/2550136832\) | \(-78648717645700923392\) | \([]\) | \(1306368\) | \(2.4978\) | |
| 59584.z2 | 59584cl1 | \([0, -1, 0, -48673, 4150721]\) | \(-413493625/152\) | \(-4687828877312\) | \([]\) | \(145152\) | \(1.3992\) | \(\Gamma_0(N)\)-optimal |
| 59584.z3 | 59584cl2 | \([0, -1, 0, 29727, 15700609]\) | \(94196375/3511808\) | \(-108307598381416448\) | \([]\) | \(435456\) | \(1.9485\) |
Rank
sage: E.rank()
The elliptic curves in class 59584.z have rank \(1\).
Complex multiplication
The elliptic curves in class 59584.z do not have complex multiplication.Modular form 59584.2.a.z
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
