Show commands:
SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 154869n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 154869.l5 | 154869n1 | \([1, 1, 0, -8671, -449456]\) | \(-1532808577/938223\) | \(-44139527609463\) | \([2]\) | \(442368\) | \(1.3202\) | \(\Gamma_0(N)\)-optimal |
| 154869.l4 | 154869n2 | \([1, 1, 0, -154876, -23520605]\) | \(8732907467857/1656369\) | \(77925338866089\) | \([2, 2]\) | \(884736\) | \(1.6667\) | |
| 154869.l3 | 154869n3 | \([1, 1, 0, -171121, -18305960]\) | \(11779205551777/3763454409\) | \(177055028274739329\) | \([2, 2]\) | \(1769472\) | \(2.0133\) | |
| 154869.l1 | 154869n4 | \([1, 1, 0, -2477911, -1502364686]\) | \(35765103905346817/1287\) | \(60548048847\) | \([2]\) | \(1769472\) | \(2.0133\) | |
| 154869.l2 | 154869n5 | \([1, 1, 0, -1086256, 421507921]\) | \(3013001140430737/108679952667\) | \(5112944120257314627\) | \([2]\) | \(3538944\) | \(2.3599\) | |
| 154869.l6 | 154869n6 | \([1, 1, 0, 484094, -124057661]\) | \(266679605718863/296110251723\) | \(-13930767665440302963\) | \([2]\) | \(3538944\) | \(2.3599\) |
Rank
sage: E.rank()
The elliptic curves in class 154869n have rank \(0\).
Complex multiplication
The elliptic curves in class 154869n do not have complex multiplication.Modular form 154869.2.a.n
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
