# Properties

 Label 4.23616.6t13.b.a Dimension 4 Group $C_3^2:D_4$ Conductor $2^{6} \cdot 3^{2} \cdot 41$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $23616= 2^{6} \cdot 3^{2} \cdot 41$ Artin number field: Splitting field of 6.2.188928.1 defined by $f= x^{6} - 2 x^{5} + x^{4} + 2 x^{3} - 4 x^{2} + 4 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.41.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.2.188928.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$
Roots:
 $r_{ 1 }$ $=$ $9 + 18\cdot 31 + 28\cdot 31^{2} + 20\cdot 31^{3} + 18\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 + 9\cdot 31 + 5\cdot 31^{2} + 5\cdot 31^{3} + 30\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $a + 26 + \left(19 a + 18\right)\cdot 31 + \left(20 a + 5\right)\cdot 31^{2} + \left(12 a + 18\right)\cdot 31^{3} + \left(11 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $7 a + 15 + \left(12 a + 17\right)\cdot 31 + \left(11 a + 7\right)\cdot 31^{2} + \left(a + 17\right)\cdot 31^{3} + \left(2 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $30 a + 28 + \left(11 a + 24\right)\cdot 31 + \left(10 a + 27\right)\cdot 31^{2} + \left(18 a + 22\right)\cdot 31^{3} + \left(19 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 6 }$ $=$ $24 a + 29 + \left(18 a + 3\right)\cdot 31 + \left(19 a + 18\right)\cdot 31^{2} + \left(29 a + 8\right)\cdot 31^{3} + \left(28 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,3,5)(2,4,6)$ $-2$ $4$ $3$ $(1,3,5)$ $1$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,4,3,6,5,2)$ $0$ $12$ $6$ $(2,4,6)(3,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.