Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 111573.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111573.z1 | 111573bd6 | \([1, -1, 0, -41058873, -101254568864]\) | \(89254274298475942657/17457\) | \(1497219174297\) | \([2]\) | \(3145728\) | \(2.6374\) | |
| 111573.z2 | 111573bd4 | \([1, -1, 0, -2566188, -1581610325]\) | \(21790813729717297/304746849\) | \(26136955125702729\) | \([2, 2]\) | \(1572864\) | \(2.2908\) | |
| 111573.z3 | 111573bd5 | \([1, -1, 0, -2493423, -1675579046]\) | \(-19989223566735457/2584262514273\) | \(-221642171494902345033\) | \([2]\) | \(3145728\) | \(2.6374\) | |
| 111573.z4 | 111573bd3 | \([1, -1, 0, -621378, 163662169]\) | \(309368403125137/44372288367\) | \(3805639053131014407\) | \([2]\) | \(1572864\) | \(2.2908\) | |
| 111573.z5 | 111573bd2 | \([1, -1, 0, -164943, -23202320]\) | \(5786435182177/627352209\) | \(53805565466711289\) | \([2, 2]\) | \(786432\) | \(1.9442\) | |
| 111573.z6 | 111573bd1 | \([1, -1, 0, 13662, -1805441]\) | \(3288008303/18259263\) | \(-1566026159828823\) | \([2]\) | \(393216\) | \(1.5976\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 111573.z have rank \(0\).
Complex multiplication
The elliptic curves in class 111573.z do not have complex multiplication.Modular form 111573.2.a.z
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
