Basic invariants
Dimension: | $20$ |
Group: | $C_2^4 : A_5$ |
Conductor: | \(141\!\cdots\!536\)\(\medspace = 2^{26} \cdot 11^{14} \cdot 113^{14} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 10.10.152779290393664.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 40T942 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $C_2^4:A_5$ |
Projective stem field: | Galois closure of 10.10.152779290393664.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 14x^{8} + 61x^{6} - 97x^{4} + 46x^{2} - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{4} + 6x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 25 + 5\cdot 37 + 16\cdot 37^{2} + 19\cdot 37^{3} + 17\cdot 37^{4} + 7\cdot 37^{5} + 31\cdot 37^{6} + 11\cdot 37^{7} + 3\cdot 37^{8} +O(37^{10})\)
$r_{ 2 }$ |
$=$ |
\( 12 + 31\cdot 37 + 20\cdot 37^{2} + 17\cdot 37^{3} + 19\cdot 37^{4} + 29\cdot 37^{5} + 5\cdot 37^{6} + 25\cdot 37^{7} + 33\cdot 37^{8} + 36\cdot 37^{9} +O(37^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 7 a^{3} + 3 a^{2} + 20 a + 24 + \left(9 a^{3} + 5 a^{2} + 4 a + 32\right)\cdot 37 + \left(30 a^{3} + 36 a^{2} + 21 a + 23\right)\cdot 37^{2} + \left(25 a^{2} + 26 a + 18\right)\cdot 37^{3} + \left(22 a^{3} + 28 a^{2} + 20 a + 1\right)\cdot 37^{4} + \left(27 a^{3} + 33 a^{2} + 21 a + 6\right)\cdot 37^{5} + \left(15 a^{3} + 13 a^{2} + 19 a + 29\right)\cdot 37^{6} + \left(28 a^{3} + 19 a^{2} + 16 a + 14\right)\cdot 37^{7} + \left(12 a^{3} + 22 a^{2} + 26 a + 1\right)\cdot 37^{8} + \left(8 a^{3} + 12 a^{2} + 7 a + 3\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 5 a^{3} + 3 a + 16 + \left(4 a^{3} + 17 a^{2} + 9 a + 14\right)\cdot 37 + \left(9 a^{3} + 3 a^{2} + 35 a + 26\right)\cdot 37^{2} + \left(32 a^{3} + 19 a^{2} + 31 a + 8\right)\cdot 37^{3} + \left(17 a^{2} + 22 a + 31\right)\cdot 37^{4} + \left(33 a^{2} + 17 a + 26\right)\cdot 37^{5} + \left(13 a^{3} + 11 a^{2} + 25 a + 10\right)\cdot 37^{6} + \left(a^{3} + 9 a^{2} + 28 a + 15\right)\cdot 37^{7} + \left(a^{3} + a^{2} + 34 a + 22\right)\cdot 37^{8} + \left(5 a^{3} + 28 a^{2} + 18 a + 26\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 11 a^{3} + 15 a^{2} + 19 a + 21 + \left(23 a^{3} + 10 a^{2} + 8 a + 6\right)\cdot 37 + \left(15 a^{3} + 29 a^{2} + 15 a + 36\right)\cdot 37^{2} + \left(6 a^{3} + 29 a^{2} + 22 a + 19\right)\cdot 37^{3} + \left(30 a^{3} + 34 a^{2} + 33 a + 18\right)\cdot 37^{4} + \left(24 a^{3} + 29 a^{2} + 23 a + 18\right)\cdot 37^{5} + \left(27 a^{3} + 34 a^{2} + 36 a + 10\right)\cdot 37^{6} + \left(10 a^{3} + 3 a^{2} + 13 a + 20\right)\cdot 37^{7} + \left(17 a^{3} + 15 a^{2} + 30 a + 23\right)\cdot 37^{8} + \left(8 a^{3} + 30 a^{2} + 5 a + 21\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 23 a^{3} + 30 a^{2} + 9 a + 23 + \left(29 a^{3} + 22 a^{2} + 4 a + 9\right)\cdot 37 + \left(13 a^{3} + 26 a^{2} + 2 a + 32\right)\cdot 37^{2} + \left(23 a^{3} + 12 a^{2} + 13 a + 14\right)\cdot 37^{3} + \left(5 a^{3} + 23 a^{2} + 28 a + 23\right)\cdot 37^{4} + \left(33 a^{3} + 14 a^{2} + 8 a + 11\right)\cdot 37^{5} + \left(28 a^{3} + 15 a^{2} + 22 a + 11\right)\cdot 37^{6} + \left(23 a^{3} + 27 a^{2} + 13 a + 29\right)\cdot 37^{7} + \left(6 a^{3} + 18 a^{2} + 24 a + 27\right)\cdot 37^{8} + \left(31 a^{3} + 4 a^{2} + 4 a + 19\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 30 a^{3} + 34 a^{2} + 17 a + 13 + \left(27 a^{3} + 31 a^{2} + 32 a + 4\right)\cdot 37 + \left(6 a^{3} + 15 a + 13\right)\cdot 37^{2} + \left(36 a^{3} + 11 a^{2} + 10 a + 18\right)\cdot 37^{3} + \left(14 a^{3} + 8 a^{2} + 16 a + 35\right)\cdot 37^{4} + \left(9 a^{3} + 3 a^{2} + 15 a + 30\right)\cdot 37^{5} + \left(21 a^{3} + 23 a^{2} + 17 a + 7\right)\cdot 37^{6} + \left(8 a^{3} + 17 a^{2} + 20 a + 22\right)\cdot 37^{7} + \left(24 a^{3} + 14 a^{2} + 10 a + 35\right)\cdot 37^{8} + \left(28 a^{3} + 24 a^{2} + 29 a + 33\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 32 a^{3} + 34 a + 21 + \left(32 a^{3} + 20 a^{2} + 27 a + 22\right)\cdot 37 + \left(27 a^{3} + 33 a^{2} + a + 10\right)\cdot 37^{2} + \left(4 a^{3} + 17 a^{2} + 5 a + 28\right)\cdot 37^{3} + \left(36 a^{3} + 19 a^{2} + 14 a + 5\right)\cdot 37^{4} + \left(36 a^{3} + 3 a^{2} + 19 a + 10\right)\cdot 37^{5} + \left(23 a^{3} + 25 a^{2} + 11 a + 26\right)\cdot 37^{6} + \left(35 a^{3} + 27 a^{2} + 8 a + 21\right)\cdot 37^{7} + \left(35 a^{3} + 35 a^{2} + 2 a + 14\right)\cdot 37^{8} + \left(31 a^{3} + 8 a^{2} + 18 a + 10\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 26 a^{3} + 22 a^{2} + 18 a + 16 + \left(13 a^{3} + 26 a^{2} + 28 a + 30\right)\cdot 37 + \left(21 a^{3} + 7 a^{2} + 21 a\right)\cdot 37^{2} + \left(30 a^{3} + 7 a^{2} + 14 a + 17\right)\cdot 37^{3} + \left(6 a^{3} + 2 a^{2} + 3 a + 18\right)\cdot 37^{4} + \left(12 a^{3} + 7 a^{2} + 13 a + 18\right)\cdot 37^{5} + \left(9 a^{3} + 2 a^{2} + 26\right)\cdot 37^{6} + \left(26 a^{3} + 33 a^{2} + 23 a + 16\right)\cdot 37^{7} + \left(19 a^{3} + 21 a^{2} + 6 a + 13\right)\cdot 37^{8} + \left(28 a^{3} + 6 a^{2} + 31 a + 15\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 10 }$ |
$=$ |
\( 14 a^{3} + 7 a^{2} + 28 a + 14 + \left(7 a^{3} + 14 a^{2} + 32 a + 27\right)\cdot 37 + \left(23 a^{3} + 10 a^{2} + 34 a + 4\right)\cdot 37^{2} + \left(13 a^{3} + 24 a^{2} + 23 a + 22\right)\cdot 37^{3} + \left(31 a^{3} + 13 a^{2} + 8 a + 13\right)\cdot 37^{4} + \left(3 a^{3} + 22 a^{2} + 28 a + 25\right)\cdot 37^{5} + \left(8 a^{3} + 21 a^{2} + 14 a + 25\right)\cdot 37^{6} + \left(13 a^{3} + 9 a^{2} + 23 a + 7\right)\cdot 37^{7} + \left(30 a^{3} + 18 a^{2} + 12 a + 9\right)\cdot 37^{8} + \left(5 a^{3} + 32 a^{2} + 32 a + 17\right)\cdot 37^{9} +O(37^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $20$ |
$5$ | $2$ | $(1,2)(3,7)(4,8)(6,10)$ | $4$ |
$10$ | $2$ | $(1,2)(6,10)$ | $-4$ |
$60$ | $2$ | $(3,5)(4,6)(7,9)(8,10)$ | $0$ |
$80$ | $3$ | $(1,7,8)(2,3,4)$ | $-1$ |
$60$ | $4$ | $(1,6,2,10)(3,4,7,8)$ | $0$ |
$120$ | $4$ | $(1,10)(2,6)(3,4,7,8)(5,9)$ | $0$ |
$192$ | $5$ | $(1,5,6,4,7)(2,9,10,8,3)$ | $0$ |
$192$ | $5$ | $(1,6,7,5,4)(2,10,3,9,8)$ | $0$ |
$80$ | $6$ | $(1,3,4,2,7,8)(6,10)$ | $1$ |
$80$ | $6$ | $(1,8,7,2,4,3)(6,10)$ | $1$ |
$80$ | $6$ | $(1,2)(3,7)(4,9,6)(5,10,8)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.