Basic invariants
Dimension: | $2$ |
Group: | $C_3 : C_4$ |
Conductor: | \(4661281\)\(\medspace = 17^{2} \cdot 127^{2} \) |
Frobenius-Schur indicator: | $-1$ |
Artin field: | Galois closure of 12.12.8025462320079492591473139857.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3 : C_4$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.3.274193.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - x^{11} - 188 x^{10} + 295 x^{9} + 12334 x^{8} - 25091 x^{7} - 338560 x^{6} + 834240 x^{5} + \cdots - 11466193 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{4} + 3x^{2} + 19x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a^{3} + 14 a^{2} + 5 a + 20 + \left(a^{3} + 8 a^{2} + 13\right)\cdot 23 + \left(14 a^{3} + 16 a^{2} + 3 a + 4\right)\cdot 23^{2} + \left(17 a^{3} + 4 a^{2} + 12 a + 17\right)\cdot 23^{3} + \left(11 a^{3} + 22 a^{2} + 18 a + 20\right)\cdot 23^{4} + \left(12 a^{3} + 2 a^{2} + 3 a + 19\right)\cdot 23^{5} + \left(13 a^{3} + 2 a^{2} + 9 a + 12\right)\cdot 23^{6} + \left(16 a^{3} + 5 a^{2} + 12 a + 19\right)\cdot 23^{7} + \left(6 a^{3} + 11 a^{2} + 2 a + 14\right)\cdot 23^{8} + \left(19 a^{3} + 12 a^{2} + 21 a + 14\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 2 }$ | $=$ | \( 5 a^{3} + 9 a^{2} + a + 10 + \left(11 a^{3} + 15 a + 21\right)\cdot 23 + \left(19 a^{3} + 18 a^{2} + 16 a + 3\right)\cdot 23^{2} + \left(6 a^{3} + 4 a^{2} + 6 a + 19\right)\cdot 23^{3} + \left(21 a^{3} + 13 a^{2} + 17 a + 10\right)\cdot 23^{4} + \left(8 a^{3} + a^{2} + 18 a + 18\right)\cdot 23^{5} + \left(21 a^{3} + 17 a^{2} + 8 a + 14\right)\cdot 23^{6} + \left(2 a^{3} + 5 a^{2} + 10 a + 15\right)\cdot 23^{7} + \left(6 a^{3} + 17 a + 18\right)\cdot 23^{8} + \left(12 a^{3} + 6 a^{2} + 10 a + 19\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 3 }$ | $=$ | \( 8 a^{3} + 17 a^{2} + 2 a + 19 + \left(7 a^{3} + 10 a^{2} + a + 7\right)\cdot 23 + \left(18 a^{3} + 22 a^{2} + 7 a + 22\right)\cdot 23^{2} + \left(4 a^{3} + 10 a^{2} + 10 a + 2\right)\cdot 23^{3} + \left(4 a^{3} + 6 a^{2} + 11 a + 16\right)\cdot 23^{4} + \left(3 a^{3} + 16 a^{2} + 22 a + 21\right)\cdot 23^{5} + \left(22 a^{3} + 6 a^{2} + 10 a + 7\right)\cdot 23^{6} + \left(21 a^{3} + 15 a^{2} + 17 a + 14\right)\cdot 23^{7} + \left(2 a^{3} + 22 a^{2} + 18 a + 18\right)\cdot 23^{8} + \left(9 a^{3} + 3 a^{2} + 2 a + 9\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 4 }$ | $=$ | \( 4 a^{3} + 6 a^{2} + 20 a + 20 + \left(19 a^{3} + 6 a^{2} + 22 a + 22\right)\cdot 23 + \left(11 a^{3} + 9 a^{2} + 6 a + 13\right)\cdot 23^{2} + \left(20 a^{3} + 2 a^{2} + 15 a + 20\right)\cdot 23^{3} + \left(15 a^{3} + 16 a^{2} + 14 a + 12\right)\cdot 23^{4} + \left(7 a^{3} + 16 a^{2} + 8 a\right)\cdot 23^{5} + \left(10 a^{3} + 11 a^{2} + 13 a + 16\right)\cdot 23^{6} + \left(10 a^{3} + 5 a^{2} + 8 a + 1\right)\cdot 23^{7} + \left(18 a^{3} + 3 a^{2} + a + 9\right)\cdot 23^{8} + \left(2 a^{3} + 4 a^{2} + 21 a + 14\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 5 }$ | $=$ | \( 6 a^{3} + 10 a^{2} + a + 3 + \left(10 a^{3} + 7 a^{2} + 4\right)\cdot 23 + \left(a^{3} + a^{2} + 5 a + 3\right)\cdot 23^{2} + \left(19 a^{3} + 15 a^{2} + 2 a + 11\right)\cdot 23^{3} + \left(13 a^{3} + 7 a^{2} + 5 a + 5\right)\cdot 23^{4} + \left(21 a^{3} + 21 a^{2} + 6 a + 4\right)\cdot 23^{5} + \left(11 a^{3} + 6 a^{2} + 11 a + 18\right)\cdot 23^{6} + \left(13 a^{3} + 5 a^{2} + 10 a + 22\right)\cdot 23^{7} + \left(12 a^{3} + 7 a^{2} + 13 a + 11\right)\cdot 23^{8} + \left(22 a^{3} + 8 a^{2} + 9 a\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 6 }$ | $=$ | \( 7 a^{3} + 13 a^{2} + 5 a + 16 + \left(2 a^{3} + a^{2} + 15 a + 2\right)\cdot 23 + \left(9 a^{3} + 10 a^{2} + 14 a + 16\right)\cdot 23^{2} + \left(5 a^{3} + 17 a^{2} + 16 a + 9\right)\cdot 23^{3} + \left(19 a^{3} + 4 a^{2} + 7 a + 3\right)\cdot 23^{4} + \left(22 a^{3} + 6 a^{2} + 16 a + 22\right)\cdot 23^{5} + \left(22 a^{3} + 12 a^{2} + 6 a + 16\right)\cdot 23^{6} + \left(5 a^{3} + 5 a^{2} + 12 a + 13\right)\cdot 23^{7} + \left(4 a^{2} + 6 a + 21\right)\cdot 23^{8} + \left(9 a^{3} + 10 a^{2} + 22 a + 5\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 7 }$ | $=$ | \( 10 a^{3} + 21 a^{2} + 6 a + 13 + \left(21 a^{3} + 11 a^{2} + a + 17\right)\cdot 23 + \left(7 a^{3} + 14 a^{2} + 5 a\right)\cdot 23^{2} + \left(3 a^{3} + 20 a + 9\right)\cdot 23^{3} + \left(2 a^{3} + 21 a^{2} + a + 8\right)\cdot 23^{4} + \left(17 a^{3} + 20 a^{2} + 20 a + 14\right)\cdot 23^{5} + \left(a^{2} + 8 a + 2\right)\cdot 23^{6} + \left(2 a^{3} + 15 a^{2} + 19 a + 11\right)\cdot 23^{7} + \left(20 a^{3} + 3 a^{2} + 7 a + 21\right)\cdot 23^{8} + \left(5 a^{3} + 8 a^{2} + 14 a\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 8 }$ | $=$ | \( 2 a^{2} + 11 a + 3 + \left(12 a^{3} + 2 a^{2} + 6 a + 12\right)\cdot 23 + \left(4 a^{3} + 20 a^{2} + 21 a + 6\right)\cdot 23^{2} + \left(18 a^{3} + 12 a^{2} + 6 a + 3\right)\cdot 23^{3} + \left(10 a^{3} + 12 a^{2} + 8 a + 21\right)\cdot 23^{4} + \left(7 a^{3} + 20 a^{2} + 3 a + 8\right)\cdot 23^{5} + \left(10 a^{3} + a^{2} + 19 a + 18\right)\cdot 23^{6} + \left(a^{3} + 20 a^{2} + 3 a + 21\right)\cdot 23^{7} + \left(13 a^{3} + 7 a^{2} + 18 a + 13\right)\cdot 23^{8} + \left(8 a^{3} + 19 a^{2} + 22 a + 21\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 9 }$ | $=$ | \( 4 a^{3} + 10 a^{2} + 19 a + 3 + \left(17 a^{3} + 4 a^{2} + 6 a + 3\right)\cdot 23 + \left(6 a^{3} + 4 a^{2} + 17 a + 20\right)\cdot 23^{2} + \left(15 a^{3} + 15 a^{2} + 3 a + 22\right)\cdot 23^{3} + \left(6 a^{3} + 18 a^{2} + 12 a + 5\right)\cdot 23^{4} + \left(12 a^{3} + 6 a^{2} + 21 a + 5\right)\cdot 23^{5} + \left(13 a^{3} + 15 a^{2} + 14 a + 15\right)\cdot 23^{6} + \left(7 a^{3} + 19 a^{2} + 7 a + 16\right)\cdot 23^{7} + \left(a^{3} + 15 a^{2} + 19 a + 19\right)\cdot 23^{8} + \left(2 a^{3} + 4 a^{2} + 22 a + 21\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 10 }$ | $=$ | \( 2 a^{3} + 6 a^{2} + 15 a + 9 + \left(3 a^{3} + 3 a^{2} + 6 a + 16\right)\cdot 23 + \left(17 a^{3} + 12 a^{2} + 19 a + 18\right)\cdot 23^{2} + \left(16 a^{3} + 2 a^{2} + 16 a + 16\right)\cdot 23^{3} + \left(8 a^{3} + 4 a^{2} + 21 a + 13\right)\cdot 23^{4} + \left(21 a^{3} + 2 a^{2} + 12\right)\cdot 23^{5} + \left(11 a^{3} + 20 a^{2} + 17 a + 20\right)\cdot 23^{6} + \left(4 a^{3} + 19 a^{2} + 5 a + 19\right)\cdot 23^{7} + \left(7 a^{3} + 11 a^{2} + 7 a + 16\right)\cdot 23^{8} + \left(5 a^{3} + 11 a + 7\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 11 }$ | $=$ | \( 9 a^{3} + 17 a^{2} + 9 a + 10 + \left(16 a^{3} + 2 a^{2} + 15 a + 12\right)\cdot 23 + \left(21 a^{3} + 2 a^{2} + 12 a + 17\right)\cdot 23^{2} + \left(3 a^{3} + 7 a^{2} + 3 a + 15\right)\cdot 23^{3} + \left(17 a^{3} + 19 a^{2} + 21 a + 18\right)\cdot 23^{4} + \left(13 a^{3} + 10 a^{2} + 13 a + 14\right)\cdot 23^{5} + \left(a^{3} + 7 a^{2} + 4 a + 11\right)\cdot 23^{6} + \left(9 a^{3} + 5 a^{2} + 14 a + 10\right)\cdot 23^{7} + \left(17 a^{3} + 8 a^{2} + 18 a + 1\right)\cdot 23^{8} + \left(5 a^{3} + 14 a^{2} + 10 a + 20\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 12 }$ | $=$ | \( 6 a^{3} + 13 a^{2} + 21 a + 13 + \left(16 a^{3} + 9 a^{2} + 3\right)\cdot 23 + \left(5 a^{3} + 7 a^{2} + 9 a + 10\right)\cdot 23^{2} + \left(6 a^{3} + 21 a^{2} + 12\right)\cdot 23^{3} + \left(6 a^{3} + 14 a^{2} + 21 a\right)\cdot 23^{4} + \left(12 a^{3} + 11 a^{2} + a + 18\right)\cdot 23^{5} + \left(20 a^{3} + 11 a^{2} + 13 a + 5\right)\cdot 23^{6} + \left(18 a^{3} + 15 a^{2} + 15 a + 16\right)\cdot 23^{7} + \left(8 a^{3} + 18 a^{2} + 6 a + 15\right)\cdot 23^{8} + \left(12 a^{3} + 22 a^{2} + 14 a\right)\cdot 23^{9} +O(23^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 12 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 12 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,9)(2,12)(3,6)(4,8)(5,10)(7,11)$ | $-2$ |
$2$ | $3$ | $(1,4,5)(2,6,11)(3,7,12)(8,10,9)$ | $-1$ |
$3$ | $4$ | $(1,7,9,11)(2,5,12,10)(3,8,6,4)$ | $0$ |
$3$ | $4$ | $(1,11,9,7)(2,10,12,5)(3,4,6,8)$ | $0$ |
$2$ | $6$ | $(1,10,4,9,5,8)(2,7,6,12,11,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.