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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 190608df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 190608.bh3 | 190608df1 | \([0, -1, 0, -4452, -108768]\) | \(810448/33\) | \(397443602688\) | \([2]\) | \(230400\) | \(0.99149\) | \(\Gamma_0(N)\)-optimal |
| 190608.bh2 | 190608df2 | \([0, -1, 0, -11672, 341760]\) | \(3650692/1089\) | \(52462555554816\) | \([2, 2]\) | \(460800\) | \(1.3381\) | |
| 190608.bh1 | 190608df3 | \([0, -1, 0, -170512, 27153952]\) | \(5690357426/891\) | \(85847818180608\) | \([2]\) | \(921600\) | \(1.6846\) | |
| 190608.bh4 | 190608df4 | \([0, -1, 0, 31648, 2247840]\) | \(36382894/43923\) | \(-4231979481421824\) | \([2]\) | \(921600\) | \(1.6846\) |
Rank
sage: E.rank()
The elliptic curves in class 190608df have rank \(0\).
Complex multiplication
The elliptic curves in class 190608df do not have complex multiplication.Modular form 190608.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
