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SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 76230bu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.bi6 | 76230bu1 | \([1, -1, 0, -312035004, 2121627070800]\) | \(2601656892010848045529/56330588160\) | \(72749150283570647040\) | \([2]\) | \(13271040\) | \(3.3380\) | \(\Gamma_0(N)\)-optimal |
| 76230.bi5 | 76230bu2 | \([1, -1, 0, -312383484, 2116650985488]\) | \(2610383204210122997209/12104550027662400\) | \(15632638639884053545665600\) | \([2, 2]\) | \(26542080\) | \(3.6846\) | |
| 76230.bi4 | 76230bu3 | \([1, -1, 0, -332960139, 1820863519173]\) | \(3160944030998056790089/720291785342976000\) | \(930233769104277183135744000\) | \([2]\) | \(39813120\) | \(3.8873\) | |
| 76230.bi7 | 76230bu4 | \([1, -1, 0, -153607284, 4266512488728]\) | \(-310366976336070130009/5909282337130963560\) | \(-7631648858182098795846209640\) | \([2]\) | \(53084160\) | \(4.0312\) | |
| 76230.bi3 | 76230bu5 | \([1, -1, 0, -476735364, -351684159480]\) | \(9278380528613437145689/5328033205714065000\) | \(6880984222948102720283985000\) | \([2]\) | \(53084160\) | \(4.0312\) | |
| 76230.bi2 | 76230bu6 | \([1, -1, 0, -1760334219, -26870211913275]\) | \(467116778179943012100169/28800309694464000000\) | \(37194677467680432623616000000\) | \([2, 2]\) | \(79626240\) | \(4.2339\) | |
| 76230.bi8 | 76230bu7 | \([1, -1, 0, 1375985781, -112203833737275]\) | \(223090928422700449019831/4340371122724101696000\) | \(-5605450278570745364682575424000\) | \([2]\) | \(159252480\) | \(4.5805\) | |
| 76230.bi1 | 76230bu8 | \([1, -1, 0, -27734639499, -1777782546254907]\) | \(1826870018430810435423307849/7641104625000000000\) | \(9868241870965256625000000000\) | \([2]\) | \(159252480\) | \(4.5805\) |
Rank
sage: E.rank()
The elliptic curves in class 76230bu have rank \(0\).
Complex multiplication
The elliptic curves in class 76230bu do not have complex multiplication.Modular form 76230.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
