# Properties

 Label 4.2e2_13_17e2.6t13.2c1 Dimension 4 Group $C_3^2:D_4$ Conductor $2^{2} \cdot 13 \cdot 17^{2}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $15028= 2^{2} \cdot 13 \cdot 17^{2}$ Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{4} + x^{3} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.13.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $x^{2} + 42 x + 3$
Roots:
 $r_{ 1 }$ $=$ $15 a + 16 + \left(36 a + 2\right)\cdot 43 + \left(15 a + 28\right)\cdot 43^{2} + \left(37 a + 6\right)\cdot 43^{3} + \left(16 a + 27\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $28 a + 31 + \left(6 a + 23\right)\cdot 43 + \left(27 a + 7\right)\cdot 43^{2} + \left(5 a + 28\right)\cdot 43^{3} + \left(26 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $30 + 41\cdot 43 + 28\cdot 43^{2} + 16\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $12 + 33\cdot 43 + 13\cdot 43^{2} + 25\cdot 43^{3} + 31\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 5 }$ $=$ $30 a + 27 + \left(21 a + 39\right)\cdot 43 + \left(30 a + 20\right)\cdot 43^{2} + \left(28 a + 5\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 6 }$ $=$ $13 a + 14 + \left(21 a + 31\right)\cdot 43 + \left(12 a + 29\right)\cdot 43^{2} + \left(14 a + 3\right)\cdot 43^{3} + \left(36 a + 25\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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