Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 74529z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 74529.bf4 | 74529z1 | \([1, -1, 0, 35712, -5542965]\) | \(12167/39\) | \(-16145090704777671\) | \([2]\) | \(387072\) | \(1.7932\) | \(\Gamma_0(N)\)-optimal |
| 74529.bf3 | 74529z2 | \([1, -1, 0, -336933, -64793520]\) | \(10218313/1521\) | \(629658537486329169\) | \([2, 2]\) | \(774144\) | \(2.1398\) | |
| 74529.bf2 | 74529z3 | \([1, -1, 0, -1454868, 612004329]\) | \(822656953/85683\) | \(35470764278396543187\) | \([2]\) | \(1548288\) | \(2.4864\) | |
| 74529.bf1 | 74529z4 | \([1, -1, 0, -5181318, -4538098629]\) | \(37159393753/1053\) | \(435917449028997117\) | \([2]\) | \(1548288\) | \(2.4864\) |
Rank
sage: E.rank()
The elliptic curves in class 74529z have rank \(1\).
Complex multiplication
The elliptic curves in class 74529z do not have complex multiplication.Modular form 74529.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
