Basic invariants
Dimension: | $11$ |
Group: | $\PSL(2,11)$ |
Conductor: | \(152\!\cdots\!161\)\(\medspace = 43^{6} \cdot 61^{6} \cdot 521^{6} \cdot 248167^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 11.3.13228791707738111328967747217040100607847335041.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PSL(2,11)$ |
Parity: | even |
Projective image: | $\PSL(2,11)$ |
Projective field: | Galois closure of 11.3.13228791707738111328967747217040100607847335041.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{5} + x + 14 \)
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 17 + 15\cdot 17^{2} + 6\cdot 17^{3} + 11\cdot 17^{4} + 15\cdot 17^{5} + 7\cdot 17^{6} + 8\cdot 17^{8} + 11\cdot 17^{9} +O(17^{10})\) |
$r_{ 2 }$ | $=$ | \( 9 a^{2} + 11 a + 7 + \left(8 a^{4} + 14 a^{3} + 11 a^{2} + 5 a + 9\right)\cdot 17 + \left(12 a^{4} + 14 a^{3} + 10 a^{2} + 11 a + 10\right)\cdot 17^{2} + \left(6 a^{4} + 13 a^{3} + 10 a^{2} + 13 a + 14\right)\cdot 17^{3} + \left(a^{4} + 9 a^{3} + 2 a^{2} + 15 a + 14\right)\cdot 17^{4} + \left(3 a^{4} + 12 a^{3} + 12 a^{2} + 7 a + 6\right)\cdot 17^{5} + \left(3 a^{4} + 14 a^{3} + 6 a^{2} + 15 a + 15\right)\cdot 17^{6} + \left(13 a^{4} + 2 a^{3} + 13 a^{2} + 10 a + 3\right)\cdot 17^{7} + \left(10 a^{4} + 7 a^{3} + 5 a^{2} + 9 a + 3\right)\cdot 17^{8} + \left(16 a^{3} + 7 a^{2} + 7 a + 1\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 3 }$ | $=$ | \( 2 a^{4} + 6 a^{3} + 9 a^{2} + 15 a + 11 + \left(11 a^{4} + 3 a^{3} + 2 a^{2} + 13 a + 5\right)\cdot 17 + \left(15 a^{4} + 2 a^{3} + 10 a^{2} + 7 a + 12\right)\cdot 17^{2} + \left(5 a^{4} + 16 a^{3} + 14 a^{2} + 13 a + 7\right)\cdot 17^{3} + \left(11 a^{4} + 6 a^{3} + 12 a^{2} + 14 a + 13\right)\cdot 17^{4} + \left(13 a^{4} + 9 a^{3} + 13 a^{2} + a + 6\right)\cdot 17^{5} + \left(7 a^{4} + a^{3} + 2 a^{2} + 11 a + 5\right)\cdot 17^{6} + \left(8 a^{4} + 7 a^{3} + a^{2} + 2 a + 13\right)\cdot 17^{7} + \left(9 a^{3} + 7 a^{2} + a\right)\cdot 17^{8} + \left(5 a^{4} + 14 a^{3} + 8 a^{2} + 2 a + 1\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 4 }$ | $=$ | \( 5 a^{4} + 7 a^{3} + 3 a^{2} + 10 a + 10 + \left(4 a^{4} + a^{2} + 13 a + 3\right)\cdot 17 + \left(2 a^{4} + 7 a^{3} + 4 a^{2} + 11 a + 15\right)\cdot 17^{2} + \left(2 a^{4} + 12 a^{3} + a^{2} + 16 a + 14\right)\cdot 17^{3} + \left(15 a^{4} + a^{3} + 13 a^{2} + 2\right)\cdot 17^{4} + \left(5 a^{4} + 6 a^{3} + 14 a^{2} + 14\right)\cdot 17^{5} + \left(14 a^{3} + 3 a^{2} + 16 a + 2\right)\cdot 17^{6} + \left(11 a^{4} + 6 a^{3} + 5 a + 5\right)\cdot 17^{7} + \left(6 a^{4} + 11 a^{3} + 11 a^{2} + 9 a + 2\right)\cdot 17^{8} + \left(7 a^{4} + 13 a^{3} + 7 a^{2} + 6 a + 13\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 5 }$ | $=$ | \( 6 a^{4} + 11 a^{3} + 8 a^{2} + 7 a + 4 + \left(a^{4} + 9 a^{3} + 6 a + 1\right)\cdot 17 + \left(7 a^{4} + 2 a^{2} + 10 a + 2\right)\cdot 17^{2} + \left(2 a^{4} + a^{3} + 2 a^{2} + 3 a + 15\right)\cdot 17^{3} + \left(14 a^{4} + 2 a^{3} + 15 a^{2} + 8 a + 8\right)\cdot 17^{4} + \left(3 a^{4} + 13 a^{3} + 8 a^{2} + 9 a + 12\right)\cdot 17^{5} + \left(15 a^{4} + 10 a^{3} + 13 a^{2} + 15 a + 14\right)\cdot 17^{6} + \left(7 a^{4} + 4 a^{3} + 13 a + 12\right)\cdot 17^{7} + \left(13 a^{4} + 9 a^{3} + 10 a^{2} + 7\right)\cdot 17^{8} + \left(13 a^{4} + 3 a^{3} + 6 a + 11\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 6 }$ | $=$ | \( 6 a^{4} + 12 a^{3} + a^{2} + 5 a + 5 + \left(8 a^{4} + 4 a^{3} + 3 a^{2} + 15 a + 6\right)\cdot 17 + \left(3 a^{4} + 6 a^{3} + 2 a^{2} + 13 a + 3\right)\cdot 17^{2} + \left(13 a^{4} + 9 a^{3} + 14 a + 6\right)\cdot 17^{3} + \left(3 a^{4} + 4 a^{3} + 15 a^{2} + 12 a + 13\right)\cdot 17^{4} + \left(6 a^{4} + 9 a^{3} + 2 a^{2} + 5 a + 12\right)\cdot 17^{5} + \left(11 a^{4} + 12 a^{3} + 12 a^{2} + a + 11\right)\cdot 17^{6} + \left(12 a^{3} + 15 a^{2} + 6 a\right)\cdot 17^{7} + \left(7 a^{4} + 5 a^{3} + 13 a^{2} + 8 a + 7\right)\cdot 17^{8} + \left(15 a^{4} + 4 a^{3} + 4 a^{2} + 16\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 7 }$ | $=$ | \( 7 a^{4} + 11 a^{3} + 4 a^{2} + 2 a + 16 + \left(10 a^{4} + 4 a^{3} + a^{2} + a + 7\right)\cdot 17 + \left(2 a^{4} + 14 a^{3} + 2 a^{2} + 8 a + 9\right)\cdot 17^{2} + \left(9 a^{4} + 13 a^{3} + 7 a^{2} + 16 a + 16\right)\cdot 17^{3} + \left(2 a^{4} + 12 a^{3} + 8 a^{2} + 7 a + 8\right)\cdot 17^{4} + \left(7 a^{4} + 7 a^{3} + 10 a^{2} + 12 a + 13\right)\cdot 17^{5} + \left(8 a^{4} + 7 a^{3} + 9 a^{2} + 3 a + 12\right)\cdot 17^{6} + \left(6 a^{4} + 14 a^{3} + 12 a^{2} + 2 a + 8\right)\cdot 17^{7} + \left(14 a^{3} + a^{2} + 6 a + 8\right)\cdot 17^{8} + \left(11 a^{4} + 14 a^{3} + 6 a^{2} + 8 a + 9\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 8 }$ | $=$ | \( 9 a^{4} + 12 a^{3} + 13 a^{2} + 2 a + 3 + \left(a^{4} + 3 a^{3} + 14 a^{2} + 5 a + 8\right)\cdot 17 + \left(10 a^{4} + 12 a^{3} + 15 a^{2} + 13 a + 4\right)\cdot 17^{2} + \left(a^{4} + 9 a^{3} + 11 a^{2} + a + 4\right)\cdot 17^{3} + \left(10 a^{4} + 8 a^{3} + 11 a^{2} + 3 a + 2\right)\cdot 17^{4} + \left(9 a^{4} + 16 a^{3} + 12 a^{2} + a\right)\cdot 17^{5} + \left(6 a^{4} + 5 a^{3} + 4 a^{2} + 12 a + 1\right)\cdot 17^{6} + \left(14 a^{4} + 3 a^{3} + 7 a^{2} + 13 a + 1\right)\cdot 17^{7} + \left(10 a^{4} + 4 a^{2} + 14 a + 9\right)\cdot 17^{8} + \left(8 a^{4} + 2 a^{3} + 3 a^{2} + 13 a\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 9 }$ | $=$ | \( 9 a^{4} + 14 a^{3} + a^{2} + 7 a + 4 + \left(2 a^{4} + 13 a^{3} + 16 a^{2} + 2 a + 15\right)\cdot 17 + \left(12 a^{4} + 11 a^{3} + 3 a^{2} + 7 a + 6\right)\cdot 17^{2} + \left(4 a^{4} + a^{3} + 11 a + 16\right)\cdot 17^{3} + \left(11 a^{4} + 5 a^{3} + a^{2} + 6 a + 15\right)\cdot 17^{4} + \left(13 a^{3} + 7 a^{2} + 11 a + 4\right)\cdot 17^{5} + \left(4 a^{4} + 6 a^{3} + 12 a^{2} + 13 a + 9\right)\cdot 17^{6} + \left(6 a^{4} + 14 a^{3} + a^{2} + 6 a + 8\right)\cdot 17^{7} + \left(10 a^{4} + 2 a^{3} + 16 a^{2} + 16 a + 16\right)\cdot 17^{8} + \left(5 a^{4} + 14 a^{3} + 5 a^{2} + 16 a + 11\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 10 }$ | $=$ | \( 12 a^{4} + 14 a^{3} + 2 a^{2} + 9 a + 3 + \left(4 a^{4} + 13 a^{3} + 2 a^{2} + 9 a + 10\right)\cdot 17 + \left(3 a^{4} + 3 a^{3} + 15 a^{2} + 10 a + 6\right)\cdot 17^{2} + \left(12 a^{3} + 15 a^{2} + 11 a + 9\right)\cdot 17^{3} + \left(15 a^{4} + a^{3} + 6 a^{2} + 7 a + 15\right)\cdot 17^{4} + \left(16 a^{4} + 8 a^{3} + a^{2} + 13 a\right)\cdot 17^{5} + \left(6 a^{4} + 9 a^{3} + 10 a^{2} + 16 a + 15\right)\cdot 17^{6} + \left(7 a^{4} + 6 a^{3} + 7 a^{2} + 7 a + 12\right)\cdot 17^{7} + \left(5 a^{4} + 3 a^{3} + 13 a^{2} + 10 a + 5\right)\cdot 17^{8} + \left(a^{4} + a^{3} + 9 a^{2} + 15\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 11 }$ | $=$ | \( 12 a^{4} + 15 a^{3} + a^{2} + 2 + \left(15 a^{4} + 16 a^{3} + 15 a^{2} + 12 a + 16\right)\cdot 17 + \left(15 a^{4} + 11 a^{3} + a^{2} + 7 a + 15\right)\cdot 17^{2} + \left(4 a^{4} + 11 a^{3} + 4 a^{2} + 15 a + 6\right)\cdot 17^{3} + \left(14 a^{3} + 15 a^{2} + 6 a + 11\right)\cdot 17^{4} + \left(a^{4} + 5 a^{3} + 4 a + 13\right)\cdot 17^{5} + \left(4 a^{4} + a^{3} + 9 a^{2} + 13 a + 5\right)\cdot 17^{6} + \left(9 a^{4} + 12 a^{3} + 7 a^{2} + 14 a\right)\cdot 17^{7} + \left(2 a^{4} + 3 a^{3} + a^{2} + 7 a + 16\right)\cdot 17^{8} + \left(16 a^{4} + 14 a^{2} + 5 a + 9\right)\cdot 17^{9} +O(17^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 11 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 11 }$ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $11$ |
$55$ | $2$ | $(3,10)(4,11)(5,9)(7,8)$ | $-1$ |
$110$ | $3$ | $(1,6,11)(2,8,10)(3,9,4)$ | $-1$ |
$132$ | $5$ | $(2,8,11,4,7)(3,9,6,5,10)$ | $1$ |
$132$ | $5$ | $(2,11,7,8,4)(3,6,10,9,5)$ | $1$ |
$110$ | $6$ | $(1,9,5)(2,3,7,6,8,10)(4,11)$ | $-1$ |
$60$ | $11$ | $(1,6,4,10,2,7,8,3,5,9,11)$ | $0$ |
$60$ | $11$ | $(1,4,2,8,5,11,6,10,7,3,9)$ | $0$ |