Show commands:
SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 122304.dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122304.dk1 | 122304ga4 | \([0, -1, 0, -163137, -25307295]\) | \(62275269892/39\) | \(300699549696\) | \([2]\) | \(393216\) | \(1.5227\) | |
| 122304.dk2 | 122304ga2 | \([0, -1, 0, -10257, -387855]\) | \(61918288/1521\) | \(2931820609536\) | \([2, 2]\) | \(196608\) | \(1.1761\) | |
| 122304.dk3 | 122304ga1 | \([0, -1, 0, -1437, 12573]\) | \(2725888/1053\) | \(126857622528\) | \([2]\) | \(98304\) | \(0.82951\) | \(\Gamma_0(N)\)-optimal |
| 122304.dk4 | 122304ga3 | \([0, -1, 0, 1503, -1236927]\) | \(48668/85683\) | \(-660636910682112\) | \([2]\) | \(393216\) | \(1.5227\) |
Rank
sage: E.rank()
The elliptic curves in class 122304.dk have rank \(0\).
Complex multiplication
The elliptic curves in class 122304.dk do not have complex multiplication.Modular form 122304.2.a.dk
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.
