Basic invariants
Dimension: | $10$ |
Group: | $S_6$ |
Conductor: | \(30344516420400625\)\(\medspace = 5^{4} \cdot 191^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.182405.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T164 |
Parity: | even |
Projective image: | $S_6$ |
Projective field: | Galois closure of 6.2.182405.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 40 + 15\cdot 73 + 43\cdot 73^{2} + 33\cdot 73^{3} + 30\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 47 + 27\cdot 73 + 48\cdot 73^{2} + 54\cdot 73^{3} + 59\cdot 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 61 a + 56 + \left(69 a + 68\right)\cdot 73 + \left(13 a + 32\right)\cdot 73^{2} + \left(32 a + 68\right)\cdot 73^{3} + \left(58 a + 5\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 58 + 32\cdot 73 + 12\cdot 73^{3} + 31\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 71 + 2\cdot 73 + 16\cdot 73^{2} + 45\cdot 73^{3} + 15\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( 12 a + 20 + \left(3 a + 71\right)\cdot 73 + \left(59 a + 4\right)\cdot 73^{2} + \left(40 a + 5\right)\cdot 73^{3} + \left(14 a + 3\right)\cdot 73^{4} +O(73^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $10$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
$15$ | $2$ | $(1,2)$ | $2$ |
$45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
$40$ | $3$ | $(1,2,3)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)$ | $0$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |