Basic invariants
| Dimension: | $6$ |
| Group: | $\PGL(2,5)$ |
| Conductor: | \(17923019113\)\(\medspace = 2617^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.17923019113.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T30 |
| Parity: | even |
| Determinant: | 1.2617.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 6.2.17923019113.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 18x^{4} + 3x^{3} + 209x^{2} + 15x - 1064 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 23\cdot 29 + 27\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 29 + 21\cdot 29^{2} + 4\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 15 + \left(23 a + 15\right)\cdot 29 + \left(3 a + 1\right)\cdot 29^{2} + \left(11 a + 7\right)\cdot 29^{3} + \left(4 a + 20\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 2 + \left(5 a + 7\right)\cdot 29 + \left(25 a + 26\right)\cdot 29^{2} + 17 a\cdot 29^{3} + \left(24 a + 2\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + \left(24 a + 25\right)\cdot 29 + \left(25 a + 9\right)\cdot 29^{2} + \left(28 a + 22\right)\cdot 29^{3} + \left(8 a + 17\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 27 + \left(4 a + 14\right)\cdot 29 + \left(3 a + 27\right)\cdot 29^{2} + 24\cdot 29^{3} + \left(20 a + 4\right)\cdot 29^{4} +O(29^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
| $15$ | $2$ | $(2,3)(4,6)$ | $-2$ |
| $20$ | $3$ | $(1,2,5)(3,6,4)$ | $0$ |
| $30$ | $4$ | $(1,4,6,2)$ | $0$ |
| $24$ | $5$ | $(1,3,2,5,6)$ | $1$ |
| $20$ | $6$ | $(1,6,2,4,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.