Basic invariants
Dimension: | $56$ |
Group: | $A_8$ |
Conductor: | \(498\!\cdots\!664\)\(\medspace = 2^{138} \cdot 51473^{48} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.0.19501894337558159417628591379185664.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 105 |
Parity: | even |
Projective image: | $A_8$ |
Projective field: | Galois closure of 8.0.19501894337558159417628591379185664.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 59 + 21\cdot 73 + 39\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 66\cdot 73^{5} + 70\cdot 73^{6} + 68\cdot 73^{7} + 59\cdot 73^{8} + 64\cdot 73^{9} +O(73^{10})\)
$r_{ 2 }$ |
$=$ |
\( 3 + 48\cdot 73 + 66\cdot 73^{2} + 60\cdot 73^{3} + 34\cdot 73^{4} + 58\cdot 73^{5} + 71\cdot 73^{6} + 71\cdot 73^{7} + 31\cdot 73^{8} + 55\cdot 73^{9} +O(73^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 4 a + 33 + \left(69 a + 45\right)\cdot 73 + \left(47 a + 27\right)\cdot 73^{2} + \left(27 a + 47\right)\cdot 73^{3} + \left(36 a + 21\right)\cdot 73^{4} + \left(8 a + 47\right)\cdot 73^{5} + \left(15 a + 30\right)\cdot 73^{6} + \left(a + 59\right)\cdot 73^{7} + \left(68 a + 64\right)\cdot 73^{8} + \left(42 a + 13\right)\cdot 73^{9} +O(73^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 69 a + 45 + \left(3 a + 29\right)\cdot 73 + \left(25 a + 29\right)\cdot 73^{2} + \left(45 a + 9\right)\cdot 73^{3} + \left(36 a + 30\right)\cdot 73^{4} + \left(64 a + 36\right)\cdot 73^{5} + \left(57 a + 67\right)\cdot 73^{6} + \left(71 a + 47\right)\cdot 73^{7} + \left(4 a + 48\right)\cdot 73^{8} + \left(30 a + 1\right)\cdot 73^{9} +O(73^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 51 + 58\cdot 73 + 48\cdot 73^{2} + 11\cdot 73^{4} + 10\cdot 73^{5} + 70\cdot 73^{6} + 71\cdot 73^{7} + 22\cdot 73^{8} + 15\cdot 73^{9} +O(73^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 53 + 30\cdot 73 + 61\cdot 73^{2} + 60\cdot 73^{3} + 66\cdot 73^{4} + 73^{5} + 3\cdot 73^{6} + 38\cdot 73^{7} + 54\cdot 73^{8} +O(73^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 49 a + 60 + \left(38 a + 31\right)\cdot 73 + \left(12 a + 46\right)\cdot 73^{2} + 26 a\cdot 73^{3} + \left(7 a + 6\right)\cdot 73^{4} + \left(18 a + 12\right)\cdot 73^{5} + \left(50 a + 32\right)\cdot 73^{6} + \left(15 a + 41\right)\cdot 73^{7} + \left(56 a + 37\right)\cdot 73^{8} + \left(51 a + 20\right)\cdot 73^{9} +O(73^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 24 a + 61 + \left(34 a + 25\right)\cdot 73 + \left(60 a + 45\right)\cdot 73^{2} + \left(46 a + 66\right)\cdot 73^{3} + \left(65 a + 1\right)\cdot 73^{4} + \left(54 a + 59\right)\cdot 73^{5} + \left(22 a + 18\right)\cdot 73^{6} + \left(57 a + 38\right)\cdot 73^{7} + \left(16 a + 44\right)\cdot 73^{8} + \left(21 a + 46\right)\cdot 73^{9} +O(73^{10})\)
| |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $56$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $8$ |
$210$ | $2$ | $(1,2)(3,4)$ | $0$ |
$112$ | $3$ | $(1,2,3)$ | $-4$ |
$1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
$2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$1344$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
$3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
$2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
$1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
$1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |