Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 82110.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82110.bp1 | 82110bq4 | \([1, 0, 0, -180201, 17385795]\) | \(647135840104522706449/243695383641033030\) | \(243695383641033030\) | \([2]\) | \(1376256\) | \(2.0358\) | |
| 82110.bp2 | 82110bq2 | \([1, 0, 0, -79051, -8366995]\) | \(54631881784005100849/1420422278480100\) | \(1420422278480100\) | \([2, 2]\) | \(688128\) | \(1.6893\) | |
| 82110.bp3 | 82110bq1 | \([1, 0, 0, -78551, -8480295]\) | \(53601780056497828849/37688490000\) | \(37688490000\) | \([2]\) | \(344064\) | \(1.3427\) | \(\Gamma_0(N)\)-optimal |
| 82110.bp4 | 82110bq3 | \([1, 0, 0, 14099, -26866585]\) | \(309946487145592751/312030025063117830\) | \(-312030025063117830\) | \([2]\) | \(1376256\) | \(2.0358\) |
Rank
sage: E.rank()
The elliptic curves in class 82110.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 82110.bp do not have complex multiplication.Modular form 82110.2.a.bp
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.
