Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 164730x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 164730.cc4 | 164730x1 | \([1, 1, 1, -2896, 101153]\) | \(-111284641/123120\) | \(-2971817495280\) | \([2]\) | \(491520\) | \(1.0880\) | \(\Gamma_0(N)\)-optimal |
| 164730.cc3 | 164730x2 | \([1, 1, 1, -54916, 4928609]\) | \(758800078561/324900\) | \(7842296168100\) | \([2, 2]\) | \(983040\) | \(1.4345\) | |
| 164730.cc1 | 164730x3 | \([1, 1, 1, -878566, 316597769]\) | \(3107086841064961/570\) | \(13758414330\) | \([2]\) | \(1966080\) | \(1.7811\) | |
| 164730.cc2 | 164730x4 | \([1, 1, 1, -63586, 3257033]\) | \(1177918188481/488703750\) | \(11796120486183750\) | \([2]\) | \(1966080\) | \(1.7811\) |
Rank
sage: E.rank()
The elliptic curves in class 164730x have rank \(0\).
Complex multiplication
The elliptic curves in class 164730x do not have complex multiplication.Modular form 164730.2.a.x
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.
