Basic invariants
Dimension: | $10$ |
Group: | $A_6$ |
Conductor: | \(16830042327806409\)\(\medspace = 3^{20} \cdot 13^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.1108809.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_6$ |
Parity: | even |
Projective image: | $A_6$ |
Projective field: | Galois closure of 6.2.1108809.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$:
\( x^{2} + 149x + 6 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 148 a + 108 + \left(5 a + 33\right)\cdot 151 + \left(91 a + 103\right)\cdot 151^{2} + \left(100 a + 69\right)\cdot 151^{3} + \left(109 a + 82\right)\cdot 151^{4} +O(151^{5})\)
$r_{ 2 }$ |
$=$ |
\( 90 a + 21 + \left(91 a + 142\right)\cdot 151 + \left(6 a + 19\right)\cdot 151^{2} + \left(39 a + 64\right)\cdot 151^{3} + \left(35 a + 125\right)\cdot 151^{4} +O(151^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 61 a + 50 + \left(59 a + 84\right)\cdot 151 + \left(144 a + 92\right)\cdot 151^{2} + \left(111 a + 135\right)\cdot 151^{3} + \left(115 a + 5\right)\cdot 151^{4} +O(151^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 3 a + 102 + \left(145 a + 48\right)\cdot 151 + \left(59 a + 128\right)\cdot 151^{2} + \left(50 a + 28\right)\cdot 151^{3} + \left(41 a + 50\right)\cdot 151^{4} +O(151^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 101 + 14\cdot 151 + 73\cdot 151^{2} + 32\cdot 151^{3} + 81\cdot 151^{4} +O(151^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 74 + 129\cdot 151 + 35\cdot 151^{2} + 122\cdot 151^{3} + 107\cdot 151^{4} +O(151^{5})\)
| |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $10$ |
$45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
$40$ | $3$ | $(1,2,3)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$72$ | $5$ | $(1,3,4,5,2)$ | $0$ |