Properties

Label 9.770...689.20t145.a
Dimension $9$
Group $S_6$
Conductor $7.702\times 10^{25}$
Indicator $1$

Related objects

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Basic invariants

Dimension:$9$
Group:$S_6$
Conductor:\(770\!\cdots\!689\)\(\medspace = 20627^{6}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.20627.1
Galois orbit size: $1$
Smallest permutation container: 20T145
Parity: even
Projective image: $S_6$
Projective field: 6.0.20627.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: \(x^{2} + 274 x + 5\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 62 a + 226 + \left(43 a + 116\right)\cdot 277 + \left(152 a + 9\right)\cdot 277^{2} + \left(231 a + 84\right)\cdot 277^{3} + \left(175 a + 85\right)\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 230 + 135\cdot 277 + 119\cdot 277^{2} + 146\cdot 277^{3} + 55\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 118 + 184\cdot 277 + 194\cdot 277^{2} + 65\cdot 277^{3} + 153\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 41 + 146\cdot 277 + 30\cdot 277^{2} + 9\cdot 277^{3} + 24\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 215 a + 135 + \left(233 a + 184\right)\cdot 277 + \left(124 a + 145\right)\cdot 277^{2} + \left(45 a + 72\right)\cdot 277^{3} + \left(101 a + 104\right)\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 82 + 63\cdot 277 + 54\cdot 277^{2} + 176\cdot 277^{3} + 131\cdot 277^{4} +O(277^{5})\)  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
$c1$
$1$ $1$ $()$ $9$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-3$
$15$ $2$ $(1,2)$ $-3$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $0$
$40$ $3$ $(1,2,3)$ $0$
$90$ $4$ $(1,2,3,4)(5,6)$ $1$
$90$ $4$ $(1,2,3,4)$ $1$
$144$ $5$ $(1,2,3,4,5)$ $-1$
$120$ $6$ $(1,2,3,4,5,6)$ $0$
$120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.