# Properties

 Label 2.29575.12t11.b.b Dimension $2$ Group $S_3 \times C_4$ Conductor $29575$ Root number not computed Indicator $0$

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## Basic invariants

 Dimension: $2$ Group: $S_3 \times C_4$ Conductor: $$29575$$$$\medspace = 5^{2} \cdot 7 \cdot 13^{2}$$ Artin stem field: 12.4.187441217958845703125.1 Galois orbit size: $2$ Smallest permutation container: $S_3 \times C_4$ Parity: odd Determinant: 1.7.2t1.a.a Projective image: $S_3$ Projective stem field: 3.1.5915.1

## Defining polynomial

 $f(x)$ $=$ $$x^{12} - x^{11} - 7 x^{10} + 67 x^{9} - 143 x^{8} + 504 x^{7} + 1909 x^{6} + 4463 x^{5} + 5068 x^{4} - 8334 x^{3} - 5606 x^{2} + 2079 x + 1331$$  .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{4} + 3 x^{2} + 19 x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$9 a^{3} + 19 a^{2} + 7 a + 11 + \left(3 a^{3} + 9 a^{2} + 5 a + 1\right)\cdot 23 + \left(19 a^{3} + 14 a^{2} + 4 a + 15\right)\cdot 23^{2} + \left(15 a^{3} + 18 a^{2} + a + 8\right)\cdot 23^{3} + \left(6 a^{2} + a + 20\right)\cdot 23^{4} + \left(7 a^{3} + 4 a^{2} + 13 a + 3\right)\cdot 23^{5} + \left(9 a^{3} + 16 a^{2} + 9 a + 16\right)\cdot 23^{6} + \left(17 a^{3} + 7 a^{2} + 16 a + 15\right)\cdot 23^{7} + \left(19 a^{3} + 19 a^{2} + 4 a + 9\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 2 }$ $=$ $$22 a^{3} + 19 a^{2} + 16 a + 12 + \left(5 a^{3} + 8 a^{2} + 8 a + 9\right)\cdot 23 + \left(21 a^{3} + 17 a^{2} + 13 a + 19\right)\cdot 23^{2} + \left(20 a^{2} + 22 a + 2\right)\cdot 23^{3} + \left(15 a^{3} + 3 a^{2} + 9 a + 15\right)\cdot 23^{4} + \left(14 a^{3} + 20 a^{2} + 2 a + 21\right)\cdot 23^{5} + \left(10 a^{3} + 14 a^{2} + 17 a + 12\right)\cdot 23^{6} + \left(10 a^{3} + 6 a^{2} + 14 a + 16\right)\cdot 23^{7} + \left(18 a^{3} + 12 a^{2} + 18 a + 18\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 3 }$ $=$ $$12 a^{3} + 8 a^{2} + 7 a + 15 + \left(3 a^{3} + 8 a^{2} + 8 a + 11\right)\cdot 23 + \left(13 a^{3} + 7 a^{2} + 21 a + 5\right)\cdot 23^{2} + \left(10 a^{3} + 21 a^{2} + 21 a + 3\right)\cdot 23^{3} + \left(a^{3} + 16 a^{2} + 11 a + 15\right)\cdot 23^{4} + \left(9 a^{3} + 8 a + 7\right)\cdot 23^{5} + \left(a^{3} + 8 a + 5\right)\cdot 23^{6} + \left(11 a^{3} + 20 a^{2} + 22 a + 16\right)\cdot 23^{7} + \left(11 a^{3} + 17 a^{2} + 17 a\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 4 }$ $=$ $$3 a^{3} + 8 a^{2} + 11 a + 18 + \left(2 a^{3} + 3 a^{2} + 2 a + 15\right)\cdot 23 + \left(9 a^{3} + 22 a^{2} + 12 a + 2\right)\cdot 23^{2} + \left(7 a^{3} + 13 a^{2} + 7 a + 10\right)\cdot 23^{3} + \left(18 a^{3} + 19 a^{2} + 7 a + 16\right)\cdot 23^{4} + \left(3 a^{3} + 3 a + 4\right)\cdot 23^{5} + \left(15 a^{3} + 4 a^{2} + 11 a + 15\right)\cdot 23^{6} + \left(19 a^{3} + 3 a^{2} + 5 a + 21\right)\cdot 23^{7} + \left(2 a^{3} + 6 a^{2} + 21 a + 22\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 5 }$ $=$ $$16 a^{3} + 18 a^{2} + 11 a + \left(a^{3} + 6 a + 4\right)\cdot 23 + \left(5 a^{3} + 14 a^{2} + 18 a + 15\right)\cdot 23^{2} + \left(16 a^{3} + 2 a^{2} + 22 a + 1\right)\cdot 23^{3} + \left(17 a^{3} + 13 a^{2} + 5 a + 2\right)\cdot 23^{4} + \left(13 a^{3} + 10 a^{2} + 18 a\right)\cdot 23^{5} + \left(4 a^{3} + a^{2} + 14 a + 19\right)\cdot 23^{6} + \left(9 a^{3} + 2 a^{2} + 12 a + 16\right)\cdot 23^{7} + \left(17 a^{3} + 10 a^{2} + 9 a + 19\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 6 }$ $=$ $$13 a^{3} + a^{2} + 20 a + 12 + \left(11 a^{3} + 14 a^{2} + 10 a + 22\right)\cdot 23 + \left(10 a^{3} + 16 a^{2} + 19 a + 2\right)\cdot 23^{2} + \left(18 a^{3} + 16 a^{2} + 14 a + 5\right)\cdot 23^{3} + \left(17 a^{3} + a^{2} + 4 a + 5\right)\cdot 23^{4} + \left(15 a^{3} + 6 a^{2} + 7 a + 22\right)\cdot 23^{5} + \left(12 a^{3} + 13 a^{2} + a + 10\right)\cdot 23^{6} + \left(20 a^{2} + 5\right)\cdot 23^{7} + \left(4 a^{3} + 16 a + 20\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 7 }$ $=$ $$12 a^{3} + 12 a + 8 + \left(18 a^{3} + 15 a^{2} + 20 a + 6\right)\cdot 23 + \left(3 a^{3} + 15 a^{2} + a + 5\right)\cdot 23^{2} + \left(21 a^{3} + 16 a^{2} + 1\right)\cdot 23^{3} + \left(11 a^{3} + 11 a^{2} + 5 a + 21\right)\cdot 23^{4} + \left(a^{3} + 14 a^{2} + 7 a + 3\right)\cdot 23^{5} + \left(12 a^{3} + 20 a + 9\right)\cdot 23^{6} + \left(a^{3} + 14 a^{2} + 22 a + 17\right)\cdot 23^{7} + \left(5 a^{3} + 20 a^{2} + 2 a + 20\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 8 }$ $=$ $$9 a^{3} + 9 a^{2} + 16 a + 19 + \left(22 a^{3} + 20 a^{2} + 13 a + 17\right)\cdot 23 + \left(17 a^{3} + a^{2} + 21 a + 19\right)\cdot 23^{2} + \left(15 a^{3} + 8 a^{2} + 21 a + 20\right)\cdot 23^{3} + \left(15 a^{3} + 14 a^{2} + 10 a + 3\right)\cdot 23^{4} + \left(16 a^{2} + 7 a + 18\right)\cdot 23^{5} + \left(20 a^{3} + 4 a^{2} + a + 13\right)\cdot 23^{6} + \left(17 a^{3} + 22 a^{2} + 17 a + 9\right)\cdot 23^{7} + \left(3 a^{3} + 18 a^{2} + 5 a + 11\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 9 }$ $=$ $$21 a^{3} + 12 a^{2} + 21 a + 17 + \left(3 a^{3} + 10 a^{2} + 18 a + 14\right)\cdot 23 + \left(2 a^{3} + 15 a^{2} + 14 a + 10\right)\cdot 23^{2} + \left(9 a^{3} + 4 a^{2} + 22 a + 20\right)\cdot 23^{3} + \left(16 a^{3} + 7 a^{2} + 17 a + 17\right)\cdot 23^{4} + \left(19 a^{3} + 3 a^{2} + 22 a + 13\right)\cdot 23^{5} + \left(19 a^{3} + 22 a^{2} + 13 a + 2\right)\cdot 23^{6} + \left(2 a^{2} + 21 a + 12\right)\cdot 23^{7} + \left(10 a^{3} + 20 a^{2} + 7 a + 6\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 10 }$ $=$ $$21 a^{3} + 12 a^{2} + 13 a + 17 + \left(14 a^{3} + 8 a^{2} + 3 a + 1\right)\cdot 23 + \left(4 a^{2} + 1\right)\cdot 23^{2} + \left(20 a^{3} + a^{2} + 7 a + 10\right)\cdot 23^{3} + \left(22 a^{3} + 2 a^{2} + 18 a + 10\right)\cdot 23^{4} + \left(13 a^{3} + 13 a^{2} + 10 a + 4\right)\cdot 23^{5} + \left(13 a^{3} + 4 a^{2} + 15 a + 2\right)\cdot 23^{6} + \left(22 a^{3} + 7 a^{2} + 6 a + 6\right)\cdot 23^{7} + \left(12 a^{3} + 6 a^{2} + a + 16\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 11 }$ $=$ $$15 a^{3} + 14 a^{2} + 5 a + 15 + \left(18 a^{2} + 15 a + 20\right)\cdot 23 + \left(7 a^{3} + 18 a^{2} + 9 a + 15\right)\cdot 23^{2} + \left(6 a^{3} + 18 a^{2} + 17 a + 7\right)\cdot 23^{3} + \left(5 a^{3} + 19 a^{2} + 20 a + 16\right)\cdot 23^{4} + \left(3 a^{3} + 5 a^{2} + 3 a + 12\right)\cdot 23^{5} + \left(11 a^{3} + 19 a^{2} + 8 a\right)\cdot 23^{6} + \left(11 a^{3} + 15 a^{2} + 18 a + 16\right)\cdot 23^{7} + \left(11 a^{3} + a^{2} + 18 a + 22\right)\cdot 23^{8} +O(23^{9})$$ $r_{ 12 }$ $=$ $$8 a^{3} + 18 a^{2} + 22 a + 18 + \left(3 a^{3} + 19 a^{2} + 11\right)\cdot 23 + \left(5 a^{3} + 12 a^{2} + a + 1\right)\cdot 23^{2} + \left(19 a^{3} + 17 a^{2} + a\right)\cdot 23^{3} + \left(17 a^{3} + 20 a^{2} + a + 17\right)\cdot 23^{4} + \left(11 a^{3} + 18 a^{2} + 10 a + 1\right)\cdot 23^{5} + \left(7 a^{3} + 13 a^{2} + 16 a + 7\right)\cdot 23^{6} + \left(15 a^{3} + 15 a^{2} + 2 a + 7\right)\cdot 23^{7} + \left(20 a^{3} + 3 a^{2} + 13 a + 14\right)\cdot 23^{8} +O(23^{9})$$

## Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

 Cycle notation $(2,11)(3,12)(4,9)(6,10)$ $(1,11)(5,6)(7,9)(8,12)$ $(1,3,7,10)(2,8,4,5)(6,11,12,9)$ $(1,9)(2,4)(3,10)(5,12)(6,8)(7,11)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 12 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,7)(2,4)(3,10)(5,8)(6,12)(9,11)$ $-2$ $3$ $2$ $(1,9)(2,4)(3,10)(5,12)(6,8)(7,11)$ $0$ $3$ $2$ $(1,11)(5,6)(7,9)(8,12)$ $0$ $2$ $3$ $(1,11,2)(3,8,12)(4,7,9)(5,6,10)$ $-1$ $1$ $4$ $(1,8,7,5)(2,3,4,10)(6,11,12,9)$ $2 \zeta_{4}$ $1$ $4$ $(1,5,7,8)(2,10,4,3)(6,9,12,11)$ $-2 \zeta_{4}$ $3$ $4$ $(1,3,7,10)(2,8,4,5)(6,11,12,9)$ $0$ $3$ $4$ $(1,10,7,3)(2,5,4,8)(6,9,12,11)$ $0$ $2$ $6$ $(1,4,11,7,2,9)(3,6,8,10,12,5)$ $1$ $2$ $12$ $(1,6,4,8,11,10,7,12,2,5,9,3)$ $\zeta_{4}$ $2$ $12$ $(1,12,4,5,11,3,7,6,2,8,9,10)$ $-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.