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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 39984df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39984.dn5 | 39984df1 | \([0, 1, 0, -26672, 1546068]\) | \(4354703137/352512\) | \(169872114843648\) | \([2]\) | \(147456\) | \(1.4735\) | \(\Gamma_0(N)\)-optimal |
| 39984.dn4 | 39984df2 | \([0, 1, 0, -89392, -8514220]\) | \(163936758817/30338064\) | \(14619618883731456\) | \([2, 2]\) | \(294912\) | \(1.8200\) | |
| 39984.dn6 | 39984df3 | \([0, 1, 0, 177168, -49351212]\) | \(1276229915423/2927177028\) | \(-1410578227884736512\) | \([2]\) | \(589824\) | \(2.1666\) | |
| 39984.dn2 | 39984df4 | \([0, 1, 0, -1359472, -610532140]\) | \(576615941610337/27060804\) | \(13040339066044416\) | \([2, 2]\) | \(589824\) | \(2.1666\) | |
| 39984.dn3 | 39984df5 | \([0, 1, 0, -1288912, -676660972]\) | \(-491411892194497/125563633938\) | \(-60507897729727537152\) | \([2]\) | \(1179648\) | \(2.5132\) | |
| 39984.dn1 | 39984df6 | \([0, 1, 0, -21751312, -39053228908]\) | \(2361739090258884097/5202\) | \(2506793361408\) | \([2]\) | \(1179648\) | \(2.5132\) |
Rank
sage: E.rank()
The elliptic curves in class 39984df have rank \(0\).
Complex multiplication
The elliptic curves in class 39984df do not have complex multiplication.Modular form 39984.2.a.df
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.
