Show commands:
SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 338800df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 338800.df4 | 338800df1 | \([0, 0, 0, -1412675, -6839010750]\) | \(-2749884201/176619520\) | \(-20025104222126080000000\) | \([2]\) | \(17694720\) | \(2.9591\) | \(\Gamma_0(N)\)-optimal |
| 338800.df3 | 338800df2 | \([0, 0, 0, -63364675, -192880866750]\) | \(248158561089321/1859334400\) | \(210811155775897600000000\) | \([2, 2]\) | \(35389440\) | \(3.3056\) | |
| 338800.df2 | 338800df3 | \([0, 0, 0, -105956675, 99002109250]\) | \(1160306142246441/634128110000\) | \(71897384235501440000000000\) | \([2]\) | \(70778880\) | \(3.6522\) | |
| 338800.df1 | 338800df4 | \([0, 0, 0, -1012004675, -12391442626750]\) | \(1010962818911303721/57392720\) | \(6507181083898880000000\) | \([2]\) | \(70778880\) | \(3.6522\) |
Rank
sage: E.rank()
The elliptic curves in class 338800df have rank \(1\).
Complex multiplication
The elliptic curves in class 338800df do not have complex multiplication.Modular form 338800.2.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.
