# Properties

 Label 3.154508.6t11.a Dimension $3$ Group $S_4\times C_2$ Conductor $154508$ Indicator $1$

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## Basic invariants

 Dimension: $3$ Group: $S_4\times C_2$ Conductor: $$154508$$$$\medspace = 2^{2} \cdot 19^{2} \cdot 107$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.618032.1 Galois orbit size: $1$ Smallest permutation container: $S_4\times C_2$ Parity: odd Projective image: $S_4$ Projective field: Galois closure of 4.2.870124.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$2 + 18\cdot 23 + 3\cdot 23^{4} + 22\cdot 23^{5} + 10\cdot 23^{6} + 19\cdot 23^{7} + 11\cdot 23^{8} +O(23^{9})$$ 2 + 18*23 + 3*23^4 + 22*23^5 + 10*23^6 + 19*23^7 + 11*23^8+O(23^9) $r_{ 2 }$ $=$ $$17 a + \left(11 a + 1\right)\cdot 23 + \left(22 a + 22\right)\cdot 23^{2} + \left(18 a + 22\right)\cdot 23^{3} + \left(a + 14\right)\cdot 23^{4} + \left(6 a + 15\right)\cdot 23^{5} + \left(a + 18\right)\cdot 23^{6} + \left(a + 20\right)\cdot 23^{7} + \left(5 a + 15\right)\cdot 23^{8} +O(23^{9})$$ 17*a + (11*a + 1)*23 + (22*a + 22)*23^2 + (18*a + 22)*23^3 + (a + 14)*23^4 + (6*a + 15)*23^5 + (a + 18)*23^6 + (a + 20)*23^7 + (5*a + 15)*23^8+O(23^9) $r_{ 3 }$ $=$ $$6 a + 11 + \left(11 a + 7\right)\cdot 23 + 9\cdot 23^{2} + \left(4 a + 15\right)\cdot 23^{3} + \left(21 a + 22\right)\cdot 23^{4} + \left(16 a + 2\right)\cdot 23^{5} + \left(21 a + 15\right)\cdot 23^{6} + \left(21 a + 21\right)\cdot 23^{7} + \left(17 a + 1\right)\cdot 23^{8} +O(23^{9})$$ 6*a + 11 + (11*a + 7)*23 + 9*23^2 + (4*a + 15)*23^3 + (21*a + 22)*23^4 + (16*a + 2)*23^5 + (21*a + 15)*23^6 + (21*a + 21)*23^7 + (17*a + 1)*23^8+O(23^9) $r_{ 4 }$ $=$ $$2 a + 6 + \left(16 a + 22\right)\cdot 23 + \left(13 a + 15\right)\cdot 23^{2} + \left(6 a + 13\right)\cdot 23^{3} + \left(6 a + 14\right)\cdot 23^{4} + \left(5 a + 17\right)\cdot 23^{5} + 8\cdot 23^{6} + \left(21 a + 6\right)\cdot 23^{7} + \left(10 a + 3\right)\cdot 23^{8} +O(23^{9})$$ 2*a + 6 + (16*a + 22)*23 + (13*a + 15)*23^2 + (6*a + 13)*23^3 + (6*a + 14)*23^4 + (5*a + 17)*23^5 + 8*23^6 + (21*a + 6)*23^7 + (10*a + 3)*23^8+O(23^9) $r_{ 5 }$ $=$ $$18 + 13\cdot 23 + 16\cdot 23^{2} + 3\cdot 23^{3} + 16\cdot 23^{4} + 11\cdot 23^{5} + 11\cdot 23^{6} + 21\cdot 23^{7} + 8\cdot 23^{8} +O(23^{9})$$ 18 + 13*23 + 16*23^2 + 3*23^3 + 16*23^4 + 11*23^5 + 11*23^6 + 21*23^7 + 8*23^8+O(23^9) $r_{ 6 }$ $=$ $$21 a + 10 + \left(6 a + 6\right)\cdot 23 + \left(9 a + 4\right)\cdot 23^{2} + \left(16 a + 13\right)\cdot 23^{3} + \left(16 a + 20\right)\cdot 23^{4} + \left(17 a + 21\right)\cdot 23^{5} + \left(22 a + 3\right)\cdot 23^{6} + \left(a + 2\right)\cdot 23^{7} + \left(12 a + 4\right)\cdot 23^{8} +O(23^{9})$$ 21*a + 10 + (6*a + 6)*23 + (9*a + 4)*23^2 + (16*a + 13)*23^3 + (16*a + 20)*23^4 + (17*a + 21)*23^5 + (22*a + 3)*23^6 + (a + 2)*23^7 + (12*a + 4)*23^8+O(23^9)

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,4,2)(3,5,6)$ $(4,6)$ $(2,4)(3,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $3$ $1$ $2$ $(1,5)(2,3)(4,6)$ $-3$ $3$ $2$ $(1,5)$ $1$ $3$ $2$ $(1,5)(4,6)$ $-1$ $6$ $2$ $(2,4)(3,6)$ $-1$ $6$ $2$ $(1,5)(2,4)(3,6)$ $1$ $8$ $3$ $(1,4,2)(3,5,6)$ $0$ $6$ $4$ $(1,6,5,4)$ $-1$ $6$ $4$ $(1,6,5,4)(2,3)$ $1$ $8$ $6$ $(1,6,3,5,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.