Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 18496.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18496.i1 | 18496c3 | \([0, 0, 0, -1677356, -836153296]\) | \(82483294977/17\) | \(107567821094912\) | \([2]\) | \(147456\) | \(2.0797\) | |
| 18496.i2 | 18496c2 | \([0, 0, 0, -105196, -12970320]\) | \(20346417/289\) | \(1828652958613504\) | \([2, 2]\) | \(73728\) | \(1.7331\) | |
| 18496.i3 | 18496c4 | \([0, 0, 0, -12716, -34980560]\) | \(-35937/83521\) | \(-528480705039302656\) | \([2]\) | \(147456\) | \(2.0797\) | |
| 18496.i4 | 18496c1 | \([0, 0, 0, -12716, 235824]\) | \(35937/17\) | \(107567821094912\) | \([2]\) | \(36864\) | \(1.3865\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 18496.i have rank \(1\).
Complex multiplication
The elliptic curves in class 18496.i do not have complex multiplication.Modular form 18496.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
