# Properties

 Label 4.161307172017.12t34.b Dimension $4$ Group $C_3^2:D_4$ Conductor $161307172017$ Indicator $1$

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$161307172017$$$$\medspace = 3^{2} \cdot 2617^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.70659.1 Galois orbit size: $1$ Smallest permutation container: 12T34 Parity: even Projective image: $\SOPlus(4,2)$ Projective field: Galois closure of 6.0.70659.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $$x^{2} + 63x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$2 + 65\cdot 67 + 17\cdot 67^{2} + 10\cdot 67^{3} + 51\cdot 67^{4} +O(67^{5})$$ 2 + 65*67 + 17*67^2 + 10*67^3 + 51*67^4+O(67^5) $r_{ 2 }$ $=$ $$43 a + 62 + \left(24 a + 11\right)\cdot 67 + \left(25 a + 64\right)\cdot 67^{2} + \left(5 a + 29\right)\cdot 67^{3} + \left(36 a + 26\right)\cdot 67^{4} +O(67^{5})$$ 43*a + 62 + (24*a + 11)*67 + (25*a + 64)*67^2 + (5*a + 29)*67^3 + (36*a + 26)*67^4+O(67^5) $r_{ 3 }$ $=$ $$56 + 34\cdot 67 + 38\cdot 67^{2} + 10\cdot 67^{3} + 67^{4} +O(67^{5})$$ 56 + 34*67 + 38*67^2 + 10*67^3 + 67^4+O(67^5) $r_{ 4 }$ $=$ $$24 a + 33 + 42 a\cdot 67 + \left(41 a + 7\right)\cdot 67^{2} + \left(61 a + 26\right)\cdot 67^{3} + \left(30 a + 31\right)\cdot 67^{4} +O(67^{5})$$ 24*a + 33 + 42*a*67 + (41*a + 7)*67^2 + (61*a + 26)*67^3 + (30*a + 31)*67^4+O(67^5) $r_{ 5 }$ $=$ $$15 a + 28 + \left(33 a + 52\right)\cdot 67 + \left(35 a + 15\right)\cdot 67^{2} + \left(35 a + 42\right)\cdot 67^{3} + \left(59 a + 44\right)\cdot 67^{4} +O(67^{5})$$ 15*a + 28 + (33*a + 52)*67 + (35*a + 15)*67^2 + (35*a + 42)*67^3 + (59*a + 44)*67^4+O(67^5) $r_{ 6 }$ $=$ $$52 a + 21 + \left(33 a + 36\right)\cdot 67 + \left(31 a + 57\right)\cdot 67^{2} + \left(31 a + 14\right)\cdot 67^{3} + \left(7 a + 46\right)\cdot 67^{4} +O(67^{5})$$ 52*a + 21 + (33*a + 36)*67 + (31*a + 57)*67^2 + (31*a + 14)*67^3 + (7*a + 46)*67^4+O(67^5)

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

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

### Character values on conjugacy classes

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