Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 49419.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49419.d1 | 49419b4 | \([1, -1, 1, -264056, -52159944]\) | \(115714886617/1539\) | \(27080686925739\) | \([2]\) | \(221184\) | \(1.7209\) | |
49419.d2 | 49419b2 | \([1, -1, 1, -16961, -764184]\) | \(30664297/3249\) | \(57170339065449\) | \([2, 2]\) | \(110592\) | \(1.3743\) | |
49419.d3 | 49419b1 | \([1, -1, 1, -3956, 83742]\) | \(389017/57\) | \(1002988404657\) | \([2]\) | \(55296\) | \(1.0278\) | \(\Gamma_0(N)\)-optimal |
49419.d4 | 49419b3 | \([1, -1, 1, 22054, -3791748]\) | \(67419143/390963\) | \(-6879497467542363\) | \([2]\) | \(221184\) | \(1.7209\) |
Rank
sage: E.rank()
The elliptic curves in class 49419.d have rank \(0\).
Complex multiplication
The elliptic curves in class 49419.d do not have complex multiplication.Modular form 49419.2.a.d
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.