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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 268770bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 268770.bg6 | 268770bg1 | \([1, 1, 1, 17334, -4969641]\) | \(23862997439/457113600\) | \(-11033611060838400\) | \([2]\) | \(2621440\) | \(1.7587\) | \(\Gamma_0(N)\)-optimal |
| 268770.bg5 | 268770bg2 | \([1, 1, 1, -352586, -76290217]\) | \(200828550012481/12454560000\) | \(300622801364640000\) | \([2, 2]\) | \(5242880\) | \(2.1053\) | |
| 268770.bg4 | 268770bg3 | \([1, 1, 1, -1069306, 331380119]\) | \(5601911201812801/1271193750000\) | \(30683526852993750000\) | \([2]\) | \(10485760\) | \(2.4519\) | |
| 268770.bg2 | 268770bg4 | \([1, 1, 1, -5554586, -5041079017]\) | \(785209010066844481/3324675600\) | \(80249586697616400\) | \([2, 2]\) | \(10485760\) | \(2.4519\) | |
| 268770.bg3 | 268770bg5 | \([1, 1, 1, -5467886, -5205947737]\) | \(-749011598724977281/51173462246460\) | \(-1235202975942823255740\) | \([2]\) | \(20971520\) | \(2.7984\) | |
| 268770.bg1 | 268770bg6 | \([1, 1, 1, -88873286, -322518653497]\) | \(3216206300355197383681/57660\) | \(1391772228540\) | \([2]\) | \(20971520\) | \(2.7984\) |
Rank
sage: E.rank()
The elliptic curves in class 268770bg have rank \(1\).
Complex multiplication
The elliptic curves in class 268770bg do not have complex multiplication.Modular form 268770.2.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 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.
