Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1617j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1617.j3 | 1617j1 | \([1, 0, 1, -320, 2153]\) | \(30664297/297\) | \(34941753\) | \([2]\) | \(432\) | \(0.26718\) | \(\Gamma_0(N)\)-optimal |
| 1617.j2 | 1617j2 | \([1, 0, 1, -565, -1669]\) | \(169112377/88209\) | \(10377700641\) | \([2, 2]\) | \(864\) | \(0.61375\) | |
| 1617.j1 | 1617j3 | \([1, 0, 1, -7180, -234517]\) | \(347873904937/395307\) | \(46507473243\) | \([2]\) | \(1728\) | \(0.96032\) | |
| 1617.j4 | 1617j4 | \([1, 0, 1, 2130, -12449]\) | \(9090072503/5845851\) | \(-687758524299\) | \([2]\) | \(1728\) | \(0.96032\) |
Rank
sage: E.rank()
The elliptic curves in class 1617j have rank \(0\).
Complex multiplication
The elliptic curves in class 1617j do not have complex multiplication.Modular form 1617.2.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 Cremona 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 Cremona labels.
