Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1440.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1440.k1 | 1440k3 | \([0, 0, 0, -732, 7616]\) | \(14526784/15\) | \(44789760\) | \([4]\) | \(512\) | \(0.38604\) | |
| 1440.k2 | 1440k2 | \([0, 0, 0, -507, -4354]\) | \(38614472/405\) | \(151165440\) | \([2]\) | \(512\) | \(0.38604\) | |
| 1440.k3 | 1440k1 | \([0, 0, 0, -57, 56]\) | \(438976/225\) | \(10497600\) | \([2, 2]\) | \(256\) | \(0.039465\) | \(\Gamma_0(N)\)-optimal |
| 1440.k4 | 1440k4 | \([0, 0, 0, 213, 434]\) | \(2863288/1875\) | \(-699840000\) | \([2]\) | \(512\) | \(0.38604\) |
Rank
sage: E.rank()
The elliptic curves in class 1440.k have rank \(0\).
Complex multiplication
The elliptic curves in class 1440.k do not have complex multiplication.Modular form 1440.2.a.k
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.
