Show commands:
SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 16800bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16800.bm3 | 16800bt1 | \([0, 1, 0, -108158, 11105688]\) | \(139927692143296/27348890625\) | \(27348890625000000\) | \([2, 2]\) | \(110592\) | \(1.8707\) | \(\Gamma_0(N)\)-optimal |
| 16800.bm2 | 16800bt2 | \([0, 1, 0, -530033, -138659937]\) | \(257307998572864/19456203375\) | \(1245197016000000000\) | \([2]\) | \(221184\) | \(2.2173\) | |
| 16800.bm1 | 16800bt3 | \([0, 1, 0, -1639408, 807355688]\) | \(60910917333827912/3255076125\) | \(26040609000000000\) | \([4]\) | \(221184\) | \(2.2173\) | |
| 16800.bm4 | 16800bt4 | \([0, 1, 0, 222592, 66010188]\) | \(152461584507448/322998046875\) | \(-2583984375000000000\) | \([2]\) | \(221184\) | \(2.2173\) |
Rank
sage: E.rank()
The elliptic curves in class 16800bt have rank \(1\).
Complex multiplication
The elliptic curves in class 16800bt do not have complex multiplication.Modular form 16800.2.a.bt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
