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SageMath
E = EllipticCurve("dg1")
E.isogeny_class()
Elliptic curves in class 29040.dg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29040.dg1 | 29040bg4 | \([0, 1, 0, -6388840, 6213443300]\) | \(15897679904620804/2475\) | \(4489844198400\) | \([4]\) | \(491520\) | \(2.2743\) | |
| 29040.dg2 | 29040bg6 | \([0, 1, 0, -3388040, -2355457260]\) | \(1185450336504002/26043266205\) | \(94489056709419018240\) | \([2]\) | \(983040\) | \(2.6209\) | |
| 29040.dg3 | 29040bg3 | \([0, 1, 0, -459840, 65578500]\) | \(5927735656804/2401490025\) | \(4356491335863321600\) | \([2, 2]\) | \(491520\) | \(2.2743\) | |
| 29040.dg4 | 29040bg2 | \([0, 1, 0, -399340, 96965900]\) | \(15529488955216/6125625\) | \(2778091097760000\) | \([2, 2]\) | \(245760\) | \(1.9278\) | |
| 29040.dg5 | 29040bg1 | \([0, 1, 0, -21215, 1980900]\) | \(-37256083456/38671875\) | \(-1096153368750000\) | \([2]\) | \(122880\) | \(1.5812\) | \(\Gamma_0(N)\)-optimal |
| 29040.dg6 | 29040bg5 | \([0, 1, 0, 1500360, 478788660]\) | \(102949393183198/86815346805\) | \(-314979701967283415040\) | \([2]\) | \(983040\) | \(2.6209\) |
Rank
sage: E.rank()
The elliptic curves in class 29040.dg have rank \(0\).
Complex multiplication
The elliptic curves in class 29040.dg do not have complex multiplication.Modular form 29040.2.a.dg
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
