Show commands:
SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 4998bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4998.be5 | 4998bd1 | \([1, 1, 1, -1667, -24991]\) | \(4354703137/352512\) | \(41472684288\) | \([2]\) | \(6144\) | \(0.78031\) | \(\Gamma_0(N)\)-optimal |
| 4998.be4 | 4998bd2 | \([1, 1, 1, -5587, 130241]\) | \(163936758817/30338064\) | \(3569242891536\) | \([2, 2]\) | \(12288\) | \(1.1269\) | |
| 4998.be2 | 4998bd3 | \([1, 1, 1, -84967, 9497081]\) | \(576615941610337/27060804\) | \(3183676529796\) | \([2, 2]\) | \(24576\) | \(1.4735\) | |
| 4998.be6 | 4998bd4 | \([1, 1, 1, 11073, 776649]\) | \(1276229915423/2927177028\) | \(-344379450167172\) | \([2]\) | \(24576\) | \(1.4735\) | |
| 4998.be1 | 4998bd5 | \([1, 1, 1, -1359457, 609526973]\) | \(2361739090258884097/5202\) | \(612010098\) | \([2]\) | \(49152\) | \(1.8200\) | |
| 4998.be3 | 4998bd6 | \([1, 1, 1, -80557, 10532549]\) | \(-491411892194497/125563633938\) | \(-14772435969171762\) | \([2]\) | \(49152\) | \(1.8200\) |
Rank
sage: E.rank()
The elliptic curves in class 4998bd have rank \(1\).
Complex multiplication
The elliptic curves in class 4998bd do not have complex multiplication.Modular form 4998.2.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 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.
