# Properties

 Label 4.12352.6t13.b.a Dimension $4$ Group $C_3^2:D_4$ Conductor $12352$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$12352$$$$\medspace = 2^{6} \cdot 193$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.49408.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Determinant: 1.193.2t1.a.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.0.49408.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $$x^{2} + 96x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$26 a + 8 + \left(74 a + 34\right)\cdot 97 + \left(47 a + 22\right)\cdot 97^{2} + \left(83 a + 39\right)\cdot 97^{3} + \left(38 a + 10\right)\cdot 97^{4} +O(97^{5})$$ 26*a + 8 + (74*a + 34)*97 + (47*a + 22)*97^2 + (83*a + 39)*97^3 + (38*a + 10)*97^4+O(97^5) $r_{ 2 }$ $=$ $$52 a + 62 + \left(19 a + 3\right)\cdot 97 + \left(35 a + 55\right)\cdot 97^{2} + \left(55 a + 60\right)\cdot 97^{3} + \left(25 a + 29\right)\cdot 97^{4} +O(97^{5})$$ 52*a + 62 + (19*a + 3)*97 + (35*a + 55)*97^2 + (55*a + 60)*97^3 + (25*a + 29)*97^4+O(97^5) $r_{ 3 }$ $=$ $$93 + 66\cdot 97 + 79\cdot 97^{2} + 27\cdot 97^{3} + 71\cdot 97^{4} +O(97^{5})$$ 93 + 66*97 + 79*97^2 + 27*97^3 + 71*97^4+O(97^5) $r_{ 4 }$ $=$ $$77 + 35\cdot 97 + 67\cdot 97^{2} + 7\cdot 97^{3} + 20\cdot 97^{4} +O(97^{5})$$ 77 + 35*97 + 67*97^2 + 7*97^3 + 20*97^4+O(97^5) $r_{ 5 }$ $=$ $$45 a + 17 + \left(77 a + 68\right)\cdot 97 + \left(61 a + 70\right)\cdot 97^{2} + \left(41 a + 80\right)\cdot 97^{3} + \left(71 a + 96\right)\cdot 97^{4} +O(97^{5})$$ 45*a + 17 + (77*a + 68)*97 + (61*a + 70)*97^2 + (41*a + 80)*97^3 + (71*a + 96)*97^4+O(97^5) $r_{ 6 }$ $=$ $$71 a + 34 + \left(22 a + 82\right)\cdot 97 + \left(49 a + 92\right)\cdot 97^{2} + \left(13 a + 74\right)\cdot 97^{3} + \left(58 a + 62\right)\cdot 97^{4} +O(97^{5})$$ 71*a + 34 + (22*a + 82)*97 + (49*a + 92)*97^2 + (13*a + 74)*97^3 + (58*a + 62)*97^4+O(97^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,4,6)(2,3,5)$ $-2$ $4$ $3$ $(1,4,6)$ $1$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,3,4,5,6,2)$ $0$ $12$ $6$ $(1,4,6)(3,5)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.