Show commands:
SageMath
E = EllipticCurve("dt1")
E.isogeny_class()
Elliptic curves in class 289674dt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 289674.dt2 | 289674dt1 | \([1, -1, 1, -16902755, -26743343965]\) | \(11165451838341046875/572244736\) | \(27371694332328192\) | \([2]\) | \(9953280\) | \(2.6270\) | \(\Gamma_0(N)\)-optimal |
| 289674.dt3 | 289674dt2 | \([1, -1, 1, -16873715, -26839838077]\) | \(-11108001800138902875/79947274872976\) | \(-3824049803973594359472\) | \([2]\) | \(19906560\) | \(2.9736\) | |
| 289674.dt1 | 289674dt3 | \([1, -1, 1, -18414650, -21673965959]\) | \(19804628171203875/5638671302656\) | \(196618411127692672892928\) | \([2]\) | \(29859840\) | \(3.1763\) | |
| 289674.dt4 | 289674dt4 | \([1, -1, 1, 48493510, -143018604935]\) | \(361682234074684125/462672528510976\) | \(-16133222269120448761049088\) | \([2]\) | \(59719680\) | \(3.5229\) |
Rank
sage: E.rank()
The elliptic curves in class 289674dt have rank \(1\).
Complex multiplication
The elliptic curves in class 289674dt do not have complex multiplication.Modular form 289674.2.a.dt
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
