# Properties

 Label 3.51681125.6t11.a.a Dimension 3 Group $S_4\times C_2$ Conductor $5^{3} \cdot 643^{2}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $3$ Group: $S_4\times C_2$ Conductor: $51681125= 5^{3} \cdot 643^{2}$ Artin number field: Splitting field of 6.2.51681125.1 defined by $f= x^{6} - x^{5} - 4 x^{4} - 22 x^{3} + 4 x^{2} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4\times C_2$ Parity: Even Determinant: 1.5.2t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.2.643.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $x^{2} + 60 x + 2$
Roots:
 $r_{ 1 }$ $=$ $53 a + 37 + \left(18 a + 58\right)\cdot 61 + \left(45 a + 10\right)\cdot 61^{2} + \left(41 a + 53\right)\cdot 61^{3} + \left(50 a + 1\right)\cdot 61^{4} + \left(27 a + 49\right)\cdot 61^{5} + \left(18 a + 39\right)\cdot 61^{6} + \left(20 a + 3\right)\cdot 61^{7} + \left(28 a + 54\right)\cdot 61^{8} + \left(18 a + 31\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ $r_{ 2 }$ $=$ $55 a + 24 + \left(40 a + 42\right)\cdot 61 + \left(21 a + 12\right)\cdot 61^{2} + \left(26 a + 41\right)\cdot 61^{3} + \left(48 a + 51\right)\cdot 61^{4} + \left(36 a + 41\right)\cdot 61^{5} + \left(a + 14\right)\cdot 61^{6} + \left(7 a + 48\right)\cdot 61^{7} + \left(17 a + 5\right)\cdot 61^{8} + \left(56 a + 40\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ $r_{ 3 }$ $=$ $53 + 42\cdot 61 + 44\cdot 61^{2} + 33\cdot 61^{3} + 48\cdot 61^{4} + 45\cdot 61^{5} + 36\cdot 61^{6} + 14\cdot 61^{7} + 52\cdot 61^{8} + 60\cdot 61^{9} +O\left(61^{ 10 }\right)$ $r_{ 4 }$ $=$ $8 a + 29 + \left(42 a + 24\right)\cdot 61 + \left(15 a + 37\right)\cdot 61^{2} + \left(19 a + 49\right)\cdot 61^{3} + \left(10 a + 10\right)\cdot 61^{4} + \left(33 a + 26\right)\cdot 61^{5} + \left(42 a + 30\right)\cdot 61^{6} + \left(40 a + 5\right)\cdot 61^{7} + \left(32 a + 1\right)\cdot 61^{8} + \left(42 a + 22\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ $r_{ 5 }$ $=$ $6 a + 18 + \left(20 a + 28\right)\cdot 61 + \left(39 a + 54\right)\cdot 61^{2} + \left(34 a + 45\right)\cdot 61^{3} + \left(12 a + 12\right)\cdot 61^{4} + \left(24 a + 30\right)\cdot 61^{5} + \left(59 a + 40\right)\cdot 61^{6} + \left(53 a + 53\right)\cdot 61^{7} + \left(43 a + 15\right)\cdot 61^{8} + \left(4 a + 18\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ $r_{ 6 }$ $=$ $23 + 47\cdot 61 + 22\cdot 61^{2} + 20\cdot 61^{3} + 57\cdot 61^{4} + 50\cdot 61^{5} + 20\cdot 61^{6} + 57\cdot 61^{7} + 53\cdot 61^{8} + 9\cdot 61^{9} +O\left(61^{ 10 }\right)$

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

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

### Character values on conjugacy classes

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