Basic invariants
Dimension: | $2$ |
Group: | $S_3 \times C_4$ |
Conductor: | \(700\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 7 \) |
Artin stem field: | Galois closure of 12.0.1200500000000.1 |
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: | Galois closure of 3.1.140.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 2x^{11} + 2x^{10} - x^{9} + 5x^{8} + 2x^{7} - 3x^{6} + 2x^{5} + 5x^{4} + 9x^{3} + 7x^{2} + 3x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{4} + 3x^{2} + 19x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 22 a^{3} + 14 a^{2} + 13 a + 12 + \left(21 a^{3} + 4 a^{2} + 11 a\right)\cdot 23 + \left(4 a^{3} + 16 a^{2} + 5 a + 20\right)\cdot 23^{2} + \left(5 a^{3} + 7 a^{2} + 4 a + 22\right)\cdot 23^{3} + \left(15 a^{3} + 6 a^{2} + 16\right)\cdot 23^{4} + \left(18 a^{3} + 7 a^{2} + 16 a + 1\right)\cdot 23^{5} + \left(17 a^{3} + 16 a^{2} + 20 a + 22\right)\cdot 23^{6} + \left(a^{3} + 9 a^{2} + 10 a + 10\right)\cdot 23^{7} + \left(14 a^{3} + 4 a^{2} + 8 a + 8\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 2 }$ | $=$ | \( 15 a^{3} + 2 a^{2} + 10 a + 2 + \left(5 a^{3} + 7 a^{2} + 10 a + 17\right)\cdot 23 + \left(2 a^{3} + 10 a^{2} + a + 19\right)\cdot 23^{2} + \left(6 a^{3} + 21 a^{2} + 18 a + 17\right)\cdot 23^{3} + \left(15 a^{2} + 2 a + 11\right)\cdot 23^{4} + \left(7 a^{3} + 17 a^{2} + 17 a\right)\cdot 23^{5} + \left(6 a^{3} + 15 a^{2} + 12 a + 10\right)\cdot 23^{6} + \left(18 a^{3} + 19 a^{2} + 5 a + 15\right)\cdot 23^{7} + \left(8 a^{3} + 11 a^{2} + 16 a + 7\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 3 }$ | $=$ | \( a^{3} + a^{2} + a + 21 + \left(12 a^{3} + 2 a^{2} + 22 a + 4\right)\cdot 23 + \left(5 a^{2} + 15 a + 9\right)\cdot 23^{2} + \left(14 a^{3} + 10 a^{2} + 8 a + 8\right)\cdot 23^{3} + \left(19 a^{3} + 20 a^{2} + 4 a + 14\right)\cdot 23^{4} + \left(12 a^{3} + 11 a^{2} + 13 a\right)\cdot 23^{5} + \left(11 a^{3} + 13 a^{2} + 11 a + 15\right)\cdot 23^{6} + \left(2 a^{3} + 4 a^{2} + 7 a + 13\right)\cdot 23^{7} + \left(5 a^{3} + 3 a^{2} + 15 a + 22\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 4 }$ | $=$ | \( 22 a^{3} + 8 a + 1 + \left(20 a^{3} + 9 a^{2} + 3 a + 8\right)\cdot 23 + \left(15 a^{3} + 16 a^{2} + 22 a + 22\right)\cdot 23^{2} + \left(20 a^{3} + 16 a^{2} + 10 a + 5\right)\cdot 23^{3} + \left(12 a^{3} + 7 a^{2} + 3 a + 1\right)\cdot 23^{4} + \left(4 a^{3} + 8 a^{2} + 21 a + 9\right)\cdot 23^{5} + \left(7 a^{3} + 18 a^{2} + 2 a + 9\right)\cdot 23^{6} + \left(21 a^{3} + 22 a^{2} + 2 a + 17\right)\cdot 23^{7} + \left(15 a^{3} + a^{2} + 4 a + 19\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 5 }$ | $=$ | \( 6 a^{3} + 8 a^{2} + 14 a + 20 + \left(9 a^{3} + 22 a^{2} + 6 a + 6\right)\cdot 23 + \left(9 a^{3} + 5 a^{2} + 16 a + 21\right)\cdot 23^{2} + \left(12 a^{3} + 20 a^{2} + 4 a + 14\right)\cdot 23^{3} + \left(22 a^{3} + 15 a^{2} + 13 a + 12\right)\cdot 23^{4} + \left(22 a^{3} + 4 a^{2} + 22 a + 22\right)\cdot 23^{5} + \left(13 a^{3} + 12 a^{2} + 9 a + 13\right)\cdot 23^{6} + \left(a^{3} + 12 a^{2} + 21 a + 11\right)\cdot 23^{7} + \left(a^{3} + a^{2} + 21 a + 12\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 6 }$ | $=$ | \( 5 a^{3} + 11 a^{2} + 11 a + 1 + \left(17 a^{3} + 21 a^{2} + 21 a + 10\right)\cdot 23 + \left(10 a^{3} + 12 a^{2} + 14 a + 11\right)\cdot 23^{2} + \left(16 a^{3} + 10 a^{2} + a + 3\right)\cdot 23^{3} + \left(11 a^{3} + 3 a^{2} + 7 a + 20\right)\cdot 23^{4} + \left(18 a^{3} + 15 a^{2} + a + 22\right)\cdot 23^{5} + \left(8 a^{3} + 3 a^{2} + 10 a + 6\right)\cdot 23^{6} + \left(4 a^{3} + 16 a^{2} + 12 a\right)\cdot 23^{7} + \left(18 a^{3} + a^{2} + 7 a + 17\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 7 }$ | $=$ | \( 17 a^{3} + 17 a^{2} + 17 a + 12 + \left(16 a^{3} + 7 a^{2} + 2 a + 22\right)\cdot 23 + \left(16 a^{3} + 19 a^{2} + 9 a + 1\right)\cdot 23^{2} + \left(11 a^{3} + 12 a^{2} + 5 a + 17\right)\cdot 23^{3} + \left(18 a^{3} + a^{2} + 4 a + 2\right)\cdot 23^{4} + \left(7 a^{3} + 14 a^{2} + 3 a + 10\right)\cdot 23^{5} + \left(4 a^{3} + 19 a^{2} + 6 a + 2\right)\cdot 23^{6} + \left(4 a^{3} + 19 a^{2} + 17 a + 19\right)\cdot 23^{7} + \left(13 a^{3} + 13 a + 5\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 8 }$ | $=$ | \( 17 a^{3} + 9 a^{2} + a + 18 + \left(21 a^{3} + 21 a^{2} + 19 a + 14\right)\cdot 23 + \left(14 a^{3} + 11 a^{2} + 19 a + 7\right)\cdot 23^{2} + \left(16 a^{3} + 14 a^{2} + 3 a + 8\right)\cdot 23^{3} + \left(15 a^{3} + 19 a^{2} + 16 a + 13\right)\cdot 23^{4} + \left(14 a^{3} + 8 a^{2} + a + 4\right)\cdot 23^{5} + \left(17 a^{3} + 11 a^{2} + 12 a + 21\right)\cdot 23^{6} + \left(22 a^{3} + 15 a^{2} + 14 a + 9\right)\cdot 23^{7} + \left(14 a^{3} + 17 a^{2} + 2 a + 18\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 9 }$ | $=$ | \( 18 a^{3} + 20 a^{2} + 21 a + 10 + \left(17 a^{3} + 17 a^{2} + 13 a + 18\right)\cdot 23 + \left(6 a^{3} + 11 a^{2} + 9 a + 4\right)\cdot 23^{2} + \left(10 a^{3} + 17 a^{2} + 8 a + 12\right)\cdot 23^{3} + \left(22 a^{3} + 15 a^{2} + 11 a + 7\right)\cdot 23^{4} + \left(18 a^{3} + 11 a^{2} + 15 a + 1\right)\cdot 23^{5} + \left(7 a^{3} + 12 a^{2} + 3 a + 12\right)\cdot 23^{6} + \left(14 a^{3} + 15 a^{2} + 15 a + 20\right)\cdot 23^{7} + \left(8 a^{3} + 13 a^{2} + 14 a + 7\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 10 }$ | $=$ | \( 16 a^{3} + 3 a^{2} + 3 a + 17 + \left(19 a^{3} + 22 a^{2} + 22 a + 5\right)\cdot 23 + \left(2 a^{3} + 13 a^{2} + a + 3\right)\cdot 23^{2} + \left(18 a^{3} + 20 a^{2} + 22 a + 22\right)\cdot 23^{3} + \left(a^{3} + a^{2} + 3 a + 17\right)\cdot 23^{4} + \left(3 a^{3} + 22 a^{2} + 5 a\right)\cdot 23^{5} + \left(15 a^{2} + 10 a + 6\right)\cdot 23^{6} + \left(10 a^{3} + 20 a^{2} + 2 a + 11\right)\cdot 23^{7} + \left(17 a^{3} + 13 a^{2} + 16 a + 5\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 11 }$ | $=$ | \( 15 a^{3} + 12 a^{2} + 4 a + 17 + \left(20 a^{3} + 8 a^{2} + 13 a + 8\right)\cdot 23 + \left(12 a^{3} + 7 a^{2} + 2 a\right)\cdot 23^{2} + \left(2 a^{3} + 16 a^{2} + 13 a\right)\cdot 23^{3} + \left(17 a^{3} + 2 a^{2} + 2\right)\cdot 23^{4} + \left(19 a^{3} + 11 a^{2} + 6 a + 11\right)\cdot 23^{5} + \left(14 a^{3} + 18 a + 5\right)\cdot 23^{6} + \left(6 a^{3} + 11 a^{2} + 9\right)\cdot 23^{7} + \left(6 a^{3} + 14 a^{2} + 10\right)\cdot 23^{8} +O(23^{9})\) |
$r_{ 12 }$ | $=$ | \( 7 a^{3} + 18 a^{2} + 12 a + 9 + \left(16 a^{2} + 14 a + 20\right)\cdot 23 + \left(17 a^{3} + 6 a^{2} + 18 a + 15\right)\cdot 23^{2} + \left(3 a^{3} + 15 a^{2} + 13 a + 4\right)\cdot 23^{3} + \left(3 a^{3} + 3 a^{2} + a + 17\right)\cdot 23^{4} + \left(12 a^{3} + 5 a^{2} + 15 a + 6\right)\cdot 23^{5} + \left(4 a^{3} + 21 a^{2} + 19 a + 13\right)\cdot 23^{6} + \left(7 a^{3} + 15 a^{2} + 4 a + 21\right)\cdot 23^{7} + \left(14 a^{3} + 6 a^{2} + 17 a + 1\right)\cdot 23^{8} +O(23^{9})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,9)(2,11)(3,6)(4,8)(5,12)(7,10)$ | $-2$ |
$3$ | $2$ | $(1,4)(2,7)(3,6)(5,12)(8,9)(10,11)$ | $0$ |
$3$ | $2$ | $(1,5)(3,7)(6,10)(9,12)$ | $0$ |
$2$ | $3$ | $(1,8,5)(2,6,10)(3,7,11)(4,12,9)$ | $-1$ |
$1$ | $4$ | $(1,10,9,7)(2,4,11,8)(3,5,6,12)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,7,9,10)(2,8,11,4)(3,12,6,5)$ | $-2 \zeta_{4}$ |
$3$ | $4$ | $(1,10,9,7)(2,12,11,5)(3,8,6,4)$ | $0$ |
$3$ | $4$ | $(1,7,9,10)(2,5,11,12)(3,4,6,8)$ | $0$ |
$2$ | $6$ | $(1,12,8,9,5,4)(2,7,6,11,10,3)$ | $1$ |
$2$ | $12$ | $(1,3,4,10,5,11,9,6,8,7,12,2)$ | $\zeta_{4}$ |
$2$ | $12$ | $(1,6,4,7,5,2,9,3,8,10,12,11)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.