Show commands:
SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 212160.er
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.er1 | 212160bk4 | \([0, 1, 0, -146881, -19975681]\) | \(10694822864826248/954268745625\) | \(31269478256640000\) | \([2]\) | \(1474560\) | \(1.9050\) | |
| 212160.er2 | 212160bk2 | \([0, 1, 0, -31961, 1836135]\) | \(881532705767104/150431501025\) | \(616167428198400\) | \([2, 2]\) | \(737280\) | \(1.5584\) | |
| 212160.er3 | 212160bk1 | \([0, 1, 0, -30516, 2041614]\) | \(49106704316297536/1905531615\) | \(121954023360\) | \([2]\) | \(368640\) | \(1.2119\) | \(\Gamma_0(N)\)-optimal |
| 212160.er4 | 212160bk3 | \([0, 1, 0, 59839, 10520415]\) | \(723135198107512/1872102004695\) | \(-61345038489845760\) | \([2]\) | \(1474560\) | \(1.9050\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.er have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.er do not have complex multiplication.Modular form 212160.2.a.er
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.
