# Properties

 Label 4.11225.6t13.a.a Dimension 4 Group $C_3^2:D_4$ Conductor $5^{2} \cdot 449$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $11225= 5^{2} \cdot 449$ Artin number field: Splitting field of 6.2.56125.1 defined by $f= x^{6} - x^{5} + x^{3} - 3 x^{2} + 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.449.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.2.56125.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{2} + 38 x + 6$
Roots:
 $r_{ 1 }$ $=$ $10 a + 19 + \left(19 a + 3\right)\cdot 41 + \left(28 a + 6\right)\cdot 41^{2} + \left(4 a + 35\right)\cdot 41^{3} + \left(36 a + 26\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $35 a + 32 + \left(25 a + 6\right)\cdot 41 + \left(30 a + 39\right)\cdot 41^{2} + \left(18 a + 20\right)\cdot 41^{3} + \left(3 a + 14\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $21 + 8\cdot 41 + 23\cdot 41^{2} + 36\cdot 41^{3} + 34\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $30 + 3\cdot 41 + 4\cdot 41^{3} + 33\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 5 }$ $=$ $31 a + 8 + \left(21 a + 10\right)\cdot 41 + \left(12 a + 31\right)\cdot 41^{2} + \left(36 a + 20\right)\cdot 41^{3} + \left(4 a + 7\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 6 }$ $=$ $6 a + 14 + \left(15 a + 8\right)\cdot 41 + \left(10 a + 23\right)\cdot 41^{2} + \left(22 a + 5\right)\cdot 41^{3} + \left(37 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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