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SageMath
E = EllipticCurve("kj1")
E.isogeny_class()
Elliptic curves in class 221760.kj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.kj1 | 221760ie4 | \([0, 0, 0, -331210092, -2320082630224]\) | \(21026497979043461623321/161783881875\) | \(30917381295144960000\) | \([2]\) | \(31457280\) | \(3.3323\) | |
| 221760.kj2 | 221760ie2 | \([0, 0, 0, -20714412, -36200606416]\) | \(5143681768032498601/14238434358225\) | \(2721007179437447577600\) | \([2, 2]\) | \(15728640\) | \(2.9857\) | |
| 221760.kj3 | 221760ie3 | \([0, 0, 0, -12549612, -65028882256]\) | \(-1143792273008057401/8897444448004035\) | \(-1700328112808248348508160\) | \([2]\) | \(31457280\) | \(3.3323\) | |
| 221760.kj4 | 221760ie1 | \([0, 0, 0, -1818732, -64507984]\) | \(3481467828171481/2005331497785\) | \(383224817093250908160\) | \([2]\) | \(7864320\) | \(2.6392\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221760.kj have rank \(0\).
Complex multiplication
The elliptic curves in class 221760.kj do not have complex multiplication.Modular form 221760.2.a.kj
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.
