Properties

Label 5.32911.6t16.a
Dimension $5$
Group $S_6$
Conductor $32911$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:\(32911\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.32911.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: odd
Projective image: $S_6$
Projective field: Galois closure of 6.0.32911.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \( x^{2} + 96x + 5 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 18 a + 42 + \left(13 a + 20\right)\cdot 97 + \left(27 a + 61\right)\cdot 97^{2} + \left(84 a + 2\right)\cdot 97^{3} + \left(a + 80\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 33 a + 72 + \left(35 a + 60\right)\cdot 97 + \left(43 a + 52\right)\cdot 97^{2} + \left(61 a + 8\right)\cdot 97^{3} + \left(3 a + 63\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 64 a + 8 + \left(61 a + 63\right)\cdot 97 + \left(53 a + 60\right)\cdot 97^{2} + \left(35 a + 26\right)\cdot 97^{3} + \left(93 a + 5\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 62 a + 72 + \left(92 a + 1\right)\cdot 97 + \left(38 a + 96\right)\cdot 97^{2} + \left(96 a + 67\right)\cdot 97^{3} + \left(40 a + 51\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 79 a + 60 + \left(83 a + 15\right)\cdot 97 + \left(69 a + 75\right)\cdot 97^{2} + \left(12 a + 59\right)\cdot 97^{3} + \left(95 a + 94\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 35 a + 37 + \left(4 a + 32\right)\cdot 97 + \left(58 a + 42\right)\cdot 97^{2} + 28\cdot 97^{3} + \left(56 a + 93\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $5$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-1$
$15$ $2$ $(1,2)$ $3$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$
$40$ $3$ $(1,2,3)$ $2$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$90$ $4$ $(1,2,3,4)$ $1$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $-1$
$120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.