Basic invariants
Dimension: | $18$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(114\!\cdots\!592\)\(\medspace = 2^{12} \cdot 3^{14} \cdot 7^{9} \cdot 29^{9} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.2.57748079088.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 36T1758 |
Parity: | odd |
Projective image: | $S_4\wr C_2$ |
Projective field: | Galois closure of 8.2.57748079088.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 57 a + 55 + \left(15 a + 49\right)\cdot 59 + \left(26 a + 14\right)\cdot 59^{2} + \left(4 a + 56\right)\cdot 59^{3} + \left(8 a + 43\right)\cdot 59^{4} + \left(13 a + 35\right)\cdot 59^{5} + \left(26 a + 10\right)\cdot 59^{6} + \left(49 a + 46\right)\cdot 59^{7} + \left(a + 39\right)\cdot 59^{8} + \left(22 a + 50\right)\cdot 59^{9} +O(59^{10})\)
$r_{ 2 }$ |
$=$ |
\( 2 a + 53 + \left(43 a + 8\right)\cdot 59 + \left(32 a + 25\right)\cdot 59^{2} + \left(54 a + 34\right)\cdot 59^{3} + \left(50 a + 47\right)\cdot 59^{4} + \left(45 a + 40\right)\cdot 59^{5} + \left(32 a + 23\right)\cdot 59^{6} + \left(9 a + 10\right)\cdot 59^{7} + \left(57 a + 51\right)\cdot 59^{8} + \left(36 a + 11\right)\cdot 59^{9} +O(59^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 5 a + 13 + \left(46 a + 1\right)\cdot 59 + \left(33 a + 1\right)\cdot 59^{2} + \left(25 a + 33\right)\cdot 59^{3} + \left(37 a + 53\right)\cdot 59^{4} + \left(25 a + 44\right)\cdot 59^{5} + \left(54 a + 40\right)\cdot 59^{6} + \left(45 a + 13\right)\cdot 59^{7} + \left(8 a + 50\right)\cdot 59^{8} + \left(36 a + 17\right)\cdot 59^{9} +O(59^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 31 a + 20 + \left(2 a + 13\right)\cdot 59 + \left(52 a + 15\right)\cdot 59^{2} + \left(11 a + 29\right)\cdot 59^{3} + \left(50 a + 2\right)\cdot 59^{4} + \left(17 a + 57\right)\cdot 59^{5} + 21 a\cdot 59^{6} + \left(24 a + 3\right)\cdot 59^{7} + \left(13 a + 6\right)\cdot 59^{8} + \left(47 a + 49\right)\cdot 59^{9} +O(59^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 28 a + 51 + \left(56 a + 43\right)\cdot 59 + \left(6 a + 5\right)\cdot 59^{2} + \left(47 a + 48\right)\cdot 59^{3} + \left(8 a + 40\right)\cdot 59^{4} + \left(41 a + 24\right)\cdot 59^{5} + \left(37 a + 4\right)\cdot 59^{6} + \left(34 a + 6\right)\cdot 59^{7} + \left(45 a + 54\right)\cdot 59^{8} + \left(11 a + 23\right)\cdot 59^{9} +O(59^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 9 + 45\cdot 59 + 34\cdot 59^{2} + 37\cdot 59^{3} + 29\cdot 59^{4} + 20\cdot 59^{5} + 19\cdot 59^{6} + 2\cdot 59^{7} + 49\cdot 59^{8} + 57\cdot 59^{9} +O(59^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 18 + 31\cdot 59 + 32\cdot 59^{2} + 31\cdot 59^{3} + 11\cdot 59^{4} + 38\cdot 59^{5} + 7\cdot 59^{6} + 31\cdot 59^{7} + 31\cdot 59^{8} + 38\cdot 59^{9} +O(59^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 54 a + 18 + \left(12 a + 42\right)\cdot 59 + \left(25 a + 47\right)\cdot 59^{2} + \left(33 a + 24\right)\cdot 59^{3} + \left(21 a + 6\right)\cdot 59^{4} + \left(33 a + 33\right)\cdot 59^{5} + \left(4 a + 10\right)\cdot 59^{6} + \left(13 a + 5\right)\cdot 59^{7} + \left(50 a + 13\right)\cdot 59^{8} + \left(22 a + 45\right)\cdot 59^{9} +O(59^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $18$ |
$6$ | $2$ | $(4,6)(5,7)$ | $-6$ |
$9$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $2$ |
$12$ | $2$ | $(1,2)$ | $0$ |
$24$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $0$ |
$36$ | $2$ | $(1,2)(4,5)$ | $-2$ |
$36$ | $2$ | $(1,2)(4,6)(5,7)$ | $0$ |
$16$ | $3$ | $(1,3,8)$ | $0$ |
$64$ | $3$ | $(1,3,8)(5,6,7)$ | $0$ |
$12$ | $4$ | $(4,5,6,7)$ | $0$ |
$36$ | $4$ | $(1,2,3,8)(4,5,6,7)$ | $-2$ |
$36$ | $4$ | $(1,2,3,8)(4,6)(5,7)$ | $0$ |
$72$ | $4$ | $(1,4,3,6)(2,5,8,7)$ | $0$ |
$72$ | $4$ | $(1,2)(4,5,6,7)$ | $2$ |
$144$ | $4$ | $(1,5,2,4)(3,6)(7,8)$ | $0$ |
$48$ | $6$ | $(1,8,3)(4,6)(5,7)$ | $0$ |
$96$ | $6$ | $(1,2)(5,7,6)$ | $0$ |
$192$ | $6$ | $(1,5,3,6,8,7)(2,4)$ | $0$ |
$144$ | $8$ | $(1,4,2,5,3,6,8,7)$ | $0$ |
$96$ | $12$ | $(1,3,8)(4,5,6,7)$ | $0$ |