# Properties

 Label 4.15248.6t13.b.a Dimension $4$ Group $C_3^2:D_4$ Conductor $15248$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$15248$$$$\medspace = 2^{4} \cdot 953$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.60992.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Determinant: 1.953.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.0.60992.1

## Defining polynomial

 $f(x)$ $=$ $x^{6} + 3 x^{4} - 2 x^{3} + 4 x^{2} - 2 x + 1$.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $x^{2} + 16 x + 3$

Roots:
 $r_{ 1 }$ $=$ $8 a + 7 + 10 a\cdot 17 + \left(a + 10\right)\cdot 17^{2} + \left(10 a + 12\right)\cdot 17^{3} + \left(2 a + 16\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $10 + 2\cdot 17 + 12\cdot 17^{2} + 5\cdot 17^{3} + 14\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 3 }$ $=$ $9 a + 15 + \left(6 a + 2\right)\cdot 17 + \left(15 a + 1\right)\cdot 17^{2} + \left(6 a + 4\right)\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 4 }$ $=$ $12 a + 8 + \left(5 a + 11\right)\cdot 17 + \left(4 a + 16\right)\cdot 17^{2} + \left(6 a + 15\right)\cdot 17^{3} + \left(16 a + 10\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 5 }$ $=$ $8 + 11\cdot 17 + 12\cdot 17^{2} + 11\cdot 17^{3} + 12\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 6 }$ $=$ $5 a + 3 + \left(11 a + 5\right)\cdot 17 + \left(12 a + 15\right)\cdot 17^{2} + 10 a\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$

## 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.