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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 302016df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 302016.df5 | 302016df1 | \([0, -1, 0, -186017, -44234463]\) | \(-1532808577/938223\) | \(-435714595514744832\) | \([2]\) | \(3932160\) | \(2.0866\) | \(\Gamma_0(N)\)-optimal |
| 302016.df4 | 302016df2 | \([0, -1, 0, -3322337, -2329357215]\) | \(8732907467857/1656369\) | \(769224532822327296\) | \([2, 2]\) | \(7864320\) | \(2.4332\) | |
| 302016.df3 | 302016df3 | \([0, -1, 0, -3670817, -1810470495]\) | \(11779205551777/3763454409\) | \(1747763607964863430656\) | \([2, 2]\) | \(15728640\) | \(2.7798\) | |
| 302016.df1 | 302016df4 | \([0, -1, 0, -53154977, -149146281183]\) | \(35765103905346817/1287\) | \(597688059691008\) | \([2]\) | \(15728640\) | \(2.7798\) | |
| 302016.df2 | 302016df5 | \([0, -1, 0, -23301857, 41931412833]\) | \(3013001140430737/108679952667\) | \(50471414170046480252928\) | \([2]\) | \(31457280\) | \(3.1263\) | |
| 302016.df6 | 302016df6 | \([0, -1, 0, 10384543, -12349179423]\) | \(266679605718863/296110251723\) | \(-137514811038800177528832\) | \([2]\) | \(31457280\) | \(3.1263\) |
Rank
sage: E.rank()
The elliptic curves in class 302016df have rank \(1\).
Complex multiplication
The elliptic curves in class 302016df do not have complex multiplication.Modular form 302016.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
