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SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 301665bu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 301665.bu4 | 301665bu1 | \([1, 0, 1, -659949, 206299147]\) | \(6585576176607121/187425\) | \(904664676825\) | \([2]\) | \(2359296\) | \(1.8040\) | \(\Gamma_0(N)\)-optimal |
| 301665.bu3 | 301665bu2 | \([1, 0, 1, -660794, 205744151]\) | \(6610905152742241/35128130625\) | \(169556777053925625\) | \([2, 2]\) | \(4718592\) | \(2.1505\) | |
| 301665.bu5 | 301665bu3 | \([1, 0, 1, -301669, 428545301]\) | \(-629004249876241/16074715228425\) | \(-77589580136998845825\) | \([2]\) | \(9437184\) | \(2.4971\) | |
| 301665.bu2 | 301665bu4 | \([1, 0, 1, -1033439, -52573363]\) | \(25288177725059761/14387797265625\) | \(69447149331894140625\) | \([2, 2]\) | \(9437184\) | \(2.4971\) | |
| 301665.bu6 | 301665bu5 | \([1, 0, 1, 4094866, -417708679]\) | \(1573196002879828319/926055908203125\) | \(-4469894992218017578125\) | \([2]\) | \(18874368\) | \(2.8437\) | |
| 301665.bu1 | 301665bu6 | \([1, 0, 1, -12124064, -16218268363]\) | \(40832710302042509761/91556816413125\) | \(441927265474219468125\) | \([2]\) | \(18874368\) | \(2.8437\) |
Rank
sage: E.rank()
The elliptic curves in class 301665bu have rank \(1\).
Complex multiplication
The elliptic curves in class 301665bu do not have complex multiplication.Modular form 301665.2.a.bu
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
