Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 70785.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 70785.i1 | 70785m8 | \([1, -1, 1, -141570023, -648307965544]\) | \(242970740812818720001/24375\) | \(31479531744375\) | \([2]\) | \(3932160\) | \(2.9377\) | |
| 70785.i2 | 70785m6 | \([1, -1, 1, -8848148, -10128101794]\) | \(59319456301170001/594140625\) | \(767313586269140625\) | \([2, 2]\) | \(1966080\) | \(2.5911\) | |
| 70785.i3 | 70785m7 | \([1, -1, 1, -8635793, -10637498968]\) | \(-55150149867714721/5950927734375\) | \(-7685432554779052734375\) | \([2]\) | \(3932160\) | \(2.9377\) | |
| 70785.i4 | 70785m4 | \([1, -1, 1, -566303, -150134938]\) | \(15551989015681/1445900625\) | \(1867334343544580625\) | \([2, 2]\) | \(983040\) | \(2.2445\) | |
| 70785.i5 | 70785m2 | \([1, -1, 1, -125258, 14463056]\) | \(168288035761/27720225\) | \(35799782680973025\) | \([2, 2]\) | \(491520\) | \(1.8979\) | |
| 70785.i6 | 70785m1 | \([1, -1, 1, -119813, 15992012]\) | \(147281603041/5265\) | \(6799578856785\) | \([2]\) | \(245760\) | \(1.5514\) | \(\Gamma_0(N)\)-optimal |
| 70785.i7 | 70785m3 | \([1, -1, 1, 228667, 81142526]\) | \(1023887723039/2798036865\) | \(-3613574987228677185\) | \([2]\) | \(983040\) | \(2.2445\) | |
| 70785.i8 | 70785m5 | \([1, -1, 1, 658822, -711732238]\) | \(24487529386319/183539412225\) | \(-237035271937674521025\) | \([2]\) | \(1966080\) | \(2.5911\) |
Rank
sage: E.rank()
The elliptic curves in class 70785.i have rank \(1\).
Complex multiplication
The elliptic curves in class 70785.i do not have complex multiplication.Modular form 70785.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
