Show commands:
SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 159120.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 159120.bd1 | 159120bp4 | \([0, 0, 0, -8486643, -9515946958]\) | \(22638311752145721841/72499050\) | \(216481003315200\) | \([2]\) | \(2359296\) | \(2.3994\) | |
| 159120.bd2 | 159120bp2 | \([0, 0, 0, -530643, -148552558]\) | \(5534056064805841/9890302500\) | \(29532285020160000\) | \([2, 2]\) | \(1179648\) | \(2.0528\) | |
| 159120.bd3 | 159120bp3 | \([0, 0, 0, -362163, -244552462]\) | \(-1759334717565361/7634341406250\) | \(-22796021289600000000\) | \([2]\) | \(2359296\) | \(2.3994\) | |
| 159120.bd4 | 159120bp1 | \([0, 0, 0, -43923, -687022]\) | \(3138428376721/1747933200\) | \(5219300568268800\) | \([2]\) | \(589824\) | \(1.7063\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 159120.bd have rank \(2\).
Complex multiplication
The elliptic curves in class 159120.bd do not have complex multiplication.Modular form 159120.2.a.bd
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.
