Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1248.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1248.f1 | 1248e3 | \([0, 1, 0, -3744, 86940]\) | \(11339065490696/351\) | \(179712\) | \([2]\) | \(768\) | \(0.51272\) | |
| 1248.f2 | 1248e2 | \([0, 1, 0, -369, -513]\) | \(1360251712/771147\) | \(3158618112\) | \([2]\) | \(768\) | \(0.51272\) | |
| 1248.f3 | 1248e1 | \([0, 1, 0, -234, 1296]\) | \(22235451328/123201\) | \(7884864\) | \([2, 2]\) | \(384\) | \(0.16614\) | \(\Gamma_0(N)\)-optimal |
| 1248.f4 | 1248e4 | \([0, 1, 0, -104, 2856]\) | \(-245314376/6908733\) | \(-3537271296\) | \([4]\) | \(768\) | \(0.51272\) |
Rank
sage: E.rank()
The elliptic curves in class 1248.f have rank \(1\).
Complex multiplication
The elliptic curves in class 1248.f do not have complex multiplication.Modular form 1248.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
