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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 214890.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 214890.bg1 | 214890e7 | \([1, 0, 0, -587086706, 4905321443436]\) | \(22378473108402603447852074638369/2555436824700646696286607000\) | \(2555436824700646696286607000\) | \([2]\) | \(185131008\) | \(3.9913\) | |
| 214890.bg2 | 214890e4 | \([1, 0, 0, -569943866, 5237117287140]\) | \(20474813944470132344494131535009/40167117934971655680\) | \(40167117934971655680\) | \([6]\) | \(61710336\) | \(3.4420\) | |
| 214890.bg3 | 214890e6 | \([1, 0, 0, -141251706, -564806505564]\) | \(311677045055319490381077598369/42563644494377808321000000\) | \(42563644494377808321000000\) | \([2, 2]\) | \(92565504\) | \(3.6447\) | |
| 214890.bg4 | 214890e3 | \([1, 0, 0, -136251706, -612153505564]\) | \(279736771702225708389957598369/6524081889000000000000\) | \(6524081889000000000000\) | \([2]\) | \(46282752\) | \(3.2982\) | |
| 214890.bg5 | 214890e2 | \([1, 0, 0, -35633466, 81769961700]\) | \(5003776637992276201276341409/7001172374869514649600\) | \(7001172374869514649600\) | \([2, 6]\) | \(30855168\) | \(3.0954\) | |
| 214890.bg6 | 214890e5 | \([1, 0, 0, -25611066, 128748959460]\) | \(-1857833027106913757199899809/6086061811779278673876480\) | \(-6086061811779278673876480\) | \([6]\) | \(61710336\) | \(3.4420\) | |
| 214890.bg7 | 214890e1 | \([1, 0, 0, -2865466, 485660900]\) | \(2602013232286824367029409/1403800139323146240000\) | \(1403800139323146240000\) | \([6]\) | \(15427584\) | \(2.7489\) | \(\Gamma_0(N)\)-optimal |
| 214890.bg8 | 214890e8 | \([1, 0, 0, 224583294, -3004706454564]\) | \(1252725893447112599494399441631/4625214399848125495694607000\) | \(-4625214399848125495694607000\) | \([2]\) | \(185131008\) | \(3.9913\) |
Rank
sage: E.rank()
The elliptic curves in class 214890.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 214890.bg do not have complex multiplication.Modular form 214890.2.a.bg
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
