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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 226941.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 226941.i1 | 226941b4 | \([1, 0, 0, -3631067, -2663476398]\) | \(35765103905346817/1287\) | \(190522189143\) | \([2]\) | \(2883584\) | \(2.1088\) | |
| 226941.i2 | 226941b5 | \([1, 0, 0, -1591772, 748372053]\) | \(3013001140430737/108679952667\) | \(16088533409537265963\) | \([2]\) | \(5767168\) | \(2.4554\) | |
| 226941.i3 | 226941b3 | \([1, 0, 0, -250757, -32366880]\) | \(11779205551777/3763454409\) | \(557126319147284601\) | \([2, 2]\) | \(2883584\) | \(2.1088\) | |
| 226941.i4 | 226941b2 | \([1, 0, 0, -226952, -41627025]\) | \(8732907467857/1656369\) | \(245202057427041\) | \([2, 2]\) | \(1441792\) | \(1.7623\) | |
| 226941.i5 | 226941b1 | \([1, 0, 0, -12707, -791928]\) | \(-1532808577/938223\) | \(-138890675885247\) | \([2]\) | \(720896\) | \(1.4157\) | \(\Gamma_0(N)\)-optimal |
| 226941.i6 | 226941b6 | \([1, 0, 0, 709378, -220361313]\) | \(266679605718863/296110251723\) | \(-43834944355828086747\) | \([2]\) | \(5767168\) | \(2.4554\) |
Rank
sage: E.rank()
The elliptic curves in class 226941.i have rank \(1\).
Complex multiplication
The elliptic curves in class 226941.i do not have complex multiplication.Modular form 226941.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
