Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 11840z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11840.c3 | 11840z1 | \([0, 1, 0, -4801, 68415]\) | \(46694890801/18944000\) | \(4966055936000\) | \([2]\) | \(18432\) | \(1.1339\) | \(\Gamma_0(N)\)-optimal |
| 11840.c4 | 11840z2 | \([0, 1, 0, 15679, 514879]\) | \(1625964918479/1369000000\) | \(-358875136000000\) | \([2]\) | \(36864\) | \(1.4805\) | |
| 11840.c1 | 11840z3 | \([0, 1, 0, -337601, 75388735]\) | \(16232905099479601/4052240\) | \(1062270402560\) | \([2]\) | \(55296\) | \(1.6832\) | |
| 11840.c2 | 11840z4 | \([0, 1, 0, -336321, 75990079]\) | \(-16048965315233521/256572640900\) | \(-67258978376089600\) | \([2]\) | \(110592\) | \(2.0298\) |
Rank
sage: E.rank()
The elliptic curves in class 11840z have rank \(0\).
Complex multiplication
The elliptic curves in class 11840z do not have complex multiplication.Modular form 11840.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
