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SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 266805.bq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 266805.bq1 | 266805bq5 | \([1, -1, 1, -707034362, 7236002234486]\) | \(257260669489908001/14267882475\) | \(2167860851692561757360475\) | \([2]\) | \(94371840\) | \(3.7351\) | |
| 266805.bq2 | 266805bq3 | \([1, -1, 1, -46691987, 99550119386]\) | \(74093292126001/14707625625\) | \(2234675388562719845675625\) | \([2, 2]\) | \(47185920\) | \(3.3885\) | |
| 266805.bq3 | 266805bq2 | \([1, -1, 1, -14408582, -19653125236]\) | \(2177286259681/161417025\) | \(24525689071754113499025\) | \([2, 2]\) | \(23592960\) | \(3.0419\) | |
| 266805.bq4 | 266805bq1 | \([1, -1, 1, -14141777, -20465706544]\) | \(2058561081361/12705\) | \(1930396621153413105\) | \([2]\) | \(11796480\) | \(2.6954\) | \(\Gamma_0(N)\)-optimal |
| 266805.bq5 | 266805bq4 | \([1, -1, 1, 13605943, -86854367806]\) | \(1833318007919/22507682505\) | \(-3419815368567161673706905\) | \([2]\) | \(47185920\) | \(3.3885\) | |
| 266805.bq6 | 266805bq6 | \([1, -1, 1, 97115908, 591718259234]\) | \(666688497209279/1381398046875\) | \(-209889501940607963241796875\) | \([2]\) | \(94371840\) | \(3.7351\) |
Rank
sage: E.rank()
The elliptic curves in class 266805.bq have rank \(1\).
Complex multiplication
The elliptic curves in class 266805.bq do not have complex multiplication.Modular form 266805.2.a.bq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
