Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 129360.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.j1 | 129360k4 | \([0, -1, 0, -27474094216, 1752805928295136]\) | \(19037313645387618625546168804/82399233032965368135\) | \(9926847863905630818011765760\) | \([2]\) | \(181665792\) | \(4.5790\) | |
129360.j2 | 129360k2 | \([0, -1, 0, -1744257916, 26477999648416]\) | \(19486220601593009351102416/1221175284018082695225\) | \(36779533053297513218694662400\) | \([2, 2]\) | \(90832896\) | \(4.2325\) | |
129360.j3 | 129360k1 | \([0, -1, 0, -331881671, -1818675944910]\) | \(2147658844706816042407936/483688189481299210485\) | \(910486908868565933029596240\) | \([2]\) | \(45416448\) | \(3.8859\) | \(\Gamma_0(N)\)-optimal |
129360.j4 | 129360k3 | \([0, -1, 0, 1387558464, 111114710954640]\) | \(2452389160534358561651516/45692546768053107181875\) | \(-5504698813147832327004581760000\) | \([2]\) | \(181665792\) | \(4.5790\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.j have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.j do not have complex multiplication.Modular form 129360.2.a.j
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.