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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 162624.fj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162624.fj1 | 162624k3 | \([0, 1, 0, -311672929, 2117753608991]\) | \(7209828390823479793/49509306\) | \(22992324040239611904\) | \([4]\) | \(17694720\) | \(3.3149\) | |
| 162624.fj2 | 162624k4 | \([0, 1, 0, -27158369, 4624446495]\) | \(4770223741048753/2740574865798\) | \(1272734167814103545413632\) | \([2]\) | \(17694720\) | \(3.3149\) | |
| 162624.fj3 | 162624k2 | \([0, 1, 0, -19491809, 33041317791]\) | \(1763535241378513/4612311396\) | \(2141976261822816190464\) | \([2, 2]\) | \(8847360\) | \(2.9683\) | |
| 162624.fj4 | 162624k1 | \([0, 1, 0, -751329, 916386975]\) | \(-100999381393/723148272\) | \(-335833012867587637248\) | \([2]\) | \(4423680\) | \(2.6217\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162624.fj have rank \(0\).
Complex multiplication
The elliptic curves in class 162624.fj do not have complex multiplication.Modular form 162624.2.a.fj
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
