Basic invariants
Dimension: | $5$ |
Group: | $(C_2^4 : C_5):C_4$ |
Conductor: | \(7591385845750000\)\(\medspace = 2^{4} \cdot 5^{6} \cdot 7^{3} \cdot 97^{4} \) |
Artin stem field: | Galois closure of 10.10.531397009202500000000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | 20T88 |
Parity: | even |
Determinant: | 1.35.4t1.a.b |
Projective image: | $C_2^4:F_5$ |
Projective stem field: | Galois closure of 10.10.531397009202500000000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 125 x^{8} - 130 x^{7} + 5570 x^{6} + 9684 x^{5} - 104680 x^{4} - 225290 x^{3} + 723545 x^{2} + \cdots - 672615 \) . |
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:
$r_{ 1 }$ | $=$ | \( 20 a^{3} + 18 a^{2} + 7 a + 7 + \left(21 a^{3} + 14 a^{2} + 3 a + 4\right)\cdot 23 + \left(19 a^{3} + 6 a^{2} + 6 a + 18\right)\cdot 23^{2} + \left(14 a^{3} + 20 a^{2}\right)\cdot 23^{3} + \left(11 a^{3} + 9 a^{2} + 20 a + 11\right)\cdot 23^{4} + \left(3 a^{3} + a^{2} + 6 a + 4\right)\cdot 23^{5} + \left(13 a^{3} + 2 a^{2} + 6 a + 17\right)\cdot 23^{6} + \left(9 a^{3} + 8 a^{2} + 18 a + 9\right)\cdot 23^{7} + \left(22 a^{3} + 3 a^{2} + 10 a + 8\right)\cdot 23^{8} + \left(7 a^{3} + 15 a^{2} + 2 a + 7\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 2 }$ | $=$ | \( a^{3} + 7 a^{2} + 16 a + 13 + \left(21 a^{2} + 8 a + 13\right)\cdot 23 + \left(4 a^{3} + 2 a^{2} + 13 a + 9\right)\cdot 23^{2} + \left(2 a^{3} + 16 a^{2} + 3 a + 3\right)\cdot 23^{3} + \left(8 a^{3} + 11 a^{2} + 16 a + 9\right)\cdot 23^{4} + \left(2 a^{3} + 16 a^{2} + 11 a + 16\right)\cdot 23^{5} + \left(21 a^{3} + 20 a^{2} + 7 a + 14\right)\cdot 23^{6} + \left(a^{3} + 10 a^{2} + 14 a + 2\right)\cdot 23^{7} + \left(13 a^{3} + 9 a^{2} + 19 a + 11\right)\cdot 23^{8} + \left(21 a^{3} + 16 a^{2} + 7 a + 7\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 3 }$ | $=$ | \( 16 + 19\cdot 23 + 20\cdot 23^{2} + 13\cdot 23^{3} + 4\cdot 23^{4} + 22\cdot 23^{5} + 16\cdot 23^{6} + 11\cdot 23^{8} + 21\cdot 23^{9} +O(23^{10})\) |
$r_{ 4 }$ | $=$ | \( 13 a^{3} + 4 a + 19 + \left(a^{3} + 6 a^{2} + 17 a + 6\right)\cdot 23 + \left(13 a^{3} + 12 a^{2} + 6 a + 18\right)\cdot 23^{2} + \left(8 a^{3} + a^{2} + 11 a + 19\right)\cdot 23^{3} + \left(6 a^{3} + 17 a + 19\right)\cdot 23^{4} + \left(21 a^{3} + 20 a^{2} + 6 a + 17\right)\cdot 23^{5} + \left(5 a^{3} + 16 a^{2} + 3\right)\cdot 23^{6} + \left(9 a^{3} + 5 a^{2} + 5 a + 3\right)\cdot 23^{7} + \left(10 a^{3} + 16 a^{2} + 12\right)\cdot 23^{8} + \left(3 a^{3} + 11 a^{2} + 12 a + 2\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 5 }$ | $=$ | \( a^{3} + 16 a^{2} + a + 10 + \left(6 a^{3} + 11 a + 22\right)\cdot 23 + \left(8 a^{3} + 12 a^{2} + 20 a + 6\right)\cdot 23^{2} + \left(a^{3} + 19 a^{2} + 16 a + 7\right)\cdot 23^{3} + \left(11 a^{3} + 8 a + 7\right)\cdot 23^{4} + \left(10 a^{3} + 19 a^{2} + 7 a + 6\right)\cdot 23^{5} + \left(2 a^{3} + 13 a^{2} + 14 a + 7\right)\cdot 23^{6} + \left(7 a^{3} + 8 a^{2} + 15 a + 11\right)\cdot 23^{7} + \left(17 a^{3} + 10 a^{2} + 4 a + 15\right)\cdot 23^{8} + \left(7 a^{3} + 10 a^{2} + 14 a + 21\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 6 }$ | $=$ | \( 16 a^{3} + 4 a^{2} + 18 a + 21 + \left(19 a^{3} + 7 a^{2} + 22 a + 7\right)\cdot 23 + \left(22 a^{3} + 3 a^{2} + 12 a + 14\right)\cdot 23^{2} + \left(20 a^{3} + 21 a^{2} + 21 a + 14\right)\cdot 23^{3} + \left(4 a^{3} + 21 a^{2} + 3 a + 13\right)\cdot 23^{4} + \left(21 a^{3} + 19 a^{2} + 19 a + 8\right)\cdot 23^{5} + \left(4 a^{3} + 18 a^{2} + 5 a + 11\right)\cdot 23^{6} + \left(a^{3} + 10 a^{2} + a + 15\right)\cdot 23^{7} + \left(8 a^{3} + 8 a^{2} + 21 a + 12\right)\cdot 23^{8} + \left(5 a^{3} + 12 a^{2} + 2 a + 17\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 7 }$ | $=$ | \( 15 a^{3} + 10 a^{2} + 6 a + 5 + \left(17 a^{3} + 22 a^{2} + 20 a + 19\right)\cdot 23 + \left(8 a^{3} + 14 a^{2} + 8 a + 12\right)\cdot 23^{2} + \left(16 a^{3} + 8 a^{2} + 19 a + 9\right)\cdot 23^{3} + \left(19 a^{3} + 21 a^{2} + 6 a\right)\cdot 23^{4} + \left(19 a^{3} + a^{2} + 9 a + 5\right)\cdot 23^{5} + \left(22 a^{3} + 10 a^{2} + 3 a + 22\right)\cdot 23^{6} + \left(a^{3} + 8 a^{2} + 12\right)\cdot 23^{7} + \left(16 a^{3} + 21 a^{2} + 19 a + 14\right)\cdot 23^{8} + \left(9 a^{3} + 8 a^{2} + 22 a + 12\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 8 }$ | $=$ | \( 9 a^{3} + 17 a^{2} + 5 a + 4 + \left(4 a^{3} + 2 a^{2} + 11 a + 13\right)\cdot 23 + \left(22 a^{3} + 10 a^{2} + 13 a + 3\right)\cdot 23^{2} + \left(7 a^{3} + 11 a^{2} + 20 a + 4\right)\cdot 23^{3} + \left(21 a^{3} + 2 a^{2} + 5 a\right)\cdot 23^{4} + \left(18 a^{3} + 8 a^{2} + 8 a + 10\right)\cdot 23^{5} + \left(6 a^{3} + 4 a^{2} + 3 a + 22\right)\cdot 23^{6} + \left(10 a^{3} + 16 a^{2} + 12 a + 14\right)\cdot 23^{7} + \left(2 a^{3} + a^{2} + 17 a + 14\right)\cdot 23^{8} + \left(11 a^{3} + 2 a^{2} + 9 a + 3\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 9 }$ | $=$ | \( 6 + 16\cdot 23 + 11\cdot 23^{2} + 13\cdot 23^{3} + 15\cdot 23^{4} + 3\cdot 23^{5} + 19\cdot 23^{6} + 22\cdot 23^{7} + 8\cdot 23^{8} + 15\cdot 23^{9} +O(23^{10})\) |
$r_{ 10 }$ | $=$ | \( 17 a^{3} + 20 a^{2} + 12 a + 14 + \left(20 a^{3} + 16 a^{2} + 20 a + 14\right)\cdot 23 + \left(15 a^{3} + 6 a^{2} + 9 a + 21\right)\cdot 23^{2} + \left(19 a^{3} + 16 a^{2} + 21 a + 4\right)\cdot 23^{3} + \left(8 a^{3} + 12 a + 10\right)\cdot 23^{4} + \left(17 a^{3} + 5 a^{2} + 22 a + 20\right)\cdot 23^{5} + \left(14 a^{3} + 5 a^{2} + 4 a + 2\right)\cdot 23^{6} + \left(4 a^{3} + 2 a + 21\right)\cdot 23^{7} + \left(2 a^{3} + 21 a^{2} + 22 a + 5\right)\cdot 23^{8} + \left(2 a^{3} + 14 a^{2} + 19 a + 5\right)\cdot 23^{9} +O(23^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 10 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 10 }$ | Character value |
$1$ | $1$ | $()$ | $5$ |
$5$ | $2$ | $(1,10)(2,7)(3,9)(4,6)$ | $-3$ |
$10$ | $2$ | $(2,7)(3,9)$ | $1$ |
$20$ | $2$ | $(1,8)(2,4)(5,10)(6,7)$ | $-1$ |
$20$ | $4$ | $(1,6,10,4)(2,3,7,9)$ | $-1$ |
$40$ | $4$ | $(1,4)(2,3,7,9)(5,8)(6,10)$ | $1$ |
$40$ | $4$ | $(1,2,8,4)(5,6,10,7)$ | $\zeta_{4}$ |
$40$ | $4$ | $(1,4,8,2)(5,7,10,6)$ | $-\zeta_{4}$ |
$64$ | $5$ | $(1,5,4,3,7)(2,10,8,6,9)$ | $0$ |
$40$ | $8$ | $(1,9,6,2,10,3,4,7)(5,8)$ | $-\zeta_{4}$ |
$40$ | $8$ | $(1,2,4,9,10,7,6,3)(5,8)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.