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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 28050.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28050.df1 | 28050da4 | \([1, 0, 0, -14198688, 20591854992]\) | \(20260414982443110947641/720358602480\) | \(11255603163750000\) | \([4]\) | \(884736\) | \(2.5758\) | |
| 28050.df2 | 28050da2 | \([1, 0, 0, -888688, 320724992]\) | \(4967657717692586041/29490113030400\) | \(460783016100000000\) | \([2, 2]\) | \(442368\) | \(2.2292\) | |
| 28050.df3 | 28050da3 | \([1, 0, 0, -378688, 686394992]\) | \(-384369029857072441/12804787777021680\) | \(-200074809015963750000\) | \([2]\) | \(884736\) | \(2.5758\) | |
| 28050.df4 | 28050da1 | \([1, 0, 0, -88688, -1675008]\) | \(4937402992298041/2780405760000\) | \(43443840000000000\) | \([2]\) | \(221184\) | \(1.8826\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 28050.df have rank \(0\).
Complex multiplication
The elliptic curves in class 28050.df do not have complex multiplication.Modular form 28050.2.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 LMFDB 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 LMFDB labels.
