Show commands:
SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 68400.dj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68400.dj1 | 68400ex4 | \([0, 0, 0, -365475, 85041250]\) | \(115714886617/1539\) | \(71803584000000\) | \([2]\) | \(393216\) | \(1.8022\) | |
| 68400.dj2 | 68400ex2 | \([0, 0, 0, -23475, 1251250]\) | \(30664297/3249\) | \(151585344000000\) | \([2, 2]\) | \(196608\) | \(1.4556\) | |
| 68400.dj3 | 68400ex1 | \([0, 0, 0, -5475, -134750]\) | \(389017/57\) | \(2659392000000\) | \([2]\) | \(98304\) | \(1.1090\) | \(\Gamma_0(N)\)-optimal |
| 68400.dj4 | 68400ex3 | \([0, 0, 0, 30525, 6165250]\) | \(67419143/390963\) | \(-18240769728000000\) | \([2]\) | \(393216\) | \(1.8022\) |
Rank
sage: E.rank()
The elliptic curves in class 68400.dj have rank \(2\).
Complex multiplication
The elliptic curves in class 68400.dj do not have complex multiplication.Modular form 68400.2.a.dj
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.
