Basic invariants
Dimension: | $4$ |
Group: | $\PGL(2,5)$ |
Conductor: | \(2617\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.17923019113.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Projective image: | $S_5$ |
Projective field: | Galois closure of 6.2.17923019113.1 |
Galois action
Roots of defining polynomial
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $4$ |
$10$ | $2$ | $(1,5)(2,4)(3,6)$ | $2$ |
$15$ | $2$ | $(2,3)(4,6)$ | $0$ |
$20$ | $3$ | $(1,2,5)(3,6,4)$ | $1$ |
$30$ | $4$ | $(1,4,6,2)$ | $0$ |
$24$ | $5$ | $(1,3,2,5,6)$ | $-1$ |
$20$ | $6$ | $(1,6,2,4,5,3)$ | $-1$ |