# Properties

 Label 4.23225.6t13.a Dimension $4$ Group $C_3^2:D_4$ Conductor $23225$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$23225$$$$\medspace = 5^{2} \cdot 929$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.116125.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Projective image: $\SOPlus(4,2)$ Projective field: Galois closure of 6.2.116125.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,6)$ $2$ $9$ $2$ $(3,6)(4,5)$ $0$ $4$ $3$ $(1,4,5)(2,3,6)$ $-2$ $4$ $3$ $(1,4,5)$ $1$ $18$ $4$ $(1,2)(3,5,6,4)$ $0$ $12$ $6$ $(1,3,4,6,5,2)$ $0$ $12$ $6$ $(1,4,5)(3,6)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.