Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 72450.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.z1 | 72450bh4 | \([1, -1, 0, -110691567, 254781437341]\) | \(13167998447866683762601/5158996582031250000\) | \(58764195442199707031250000\) | \([2]\) | \(23592960\) | \(3.6433\) | |
| 72450.z2 | 72450bh2 | \([1, -1, 0, -49833567, -132579732659]\) | \(1201550658189465626281/28577902500000000\) | \(325520170664062500000000\) | \([2, 2]\) | \(11796480\) | \(3.2967\) | |
| 72450.z3 | 72450bh1 | \([1, -1, 0, -49545567, -134219316659]\) | \(1180838681727016392361/692428800000\) | \(7887196800000000000\) | \([2]\) | \(5898240\) | \(2.9501\) | \(\Gamma_0(N)\)-optimal |
| 72450.z4 | 72450bh3 | \([1, -1, 0, 6416433, -415010982659]\) | \(2564821295690373719/6533572090396050000\) | \(-74421469592167507031250000\) | \([2]\) | \(23592960\) | \(3.6433\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.z have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.z do not have complex multiplication.Modular form 72450.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
