Properties

Label 6.33792250337.20t30.a
Dimension $6$
Group $S_5$
Conductor $33792250337$
Indicator $1$

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Basic invariants

Dimension:$6$
Group:$S_5$
Conductor:\(33792250337\)\(\medspace = 53^{3} \cdot 61^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.3233.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: even
Projective image: $S_5$
Projective field: 5.1.3233.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 383 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 88 + 282\cdot 383 + 41\cdot 383^{2} + 170\cdot 383^{3} + 220\cdot 383^{4} +O(383^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 89 + 155\cdot 383 + 93\cdot 383^{2} + 241\cdot 383^{3} + 206\cdot 383^{4} +O(383^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 120 + 178\cdot 383 + 256\cdot 383^{2} + 144\cdot 383^{3} + 377\cdot 383^{4} +O(383^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 127 + 109\cdot 383 + 48\cdot 383^{2} + 292\cdot 383^{3} + 181\cdot 383^{4} +O(383^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 342 + 40\cdot 383 + 326\cdot 383^{2} + 300\cdot 383^{3} + 162\cdot 383^{4} +O(383^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character values
$c1$
$1$ $1$ $()$ $6$
$10$ $2$ $(1,2)$ $0$
$15$ $2$ $(1,2)(3,4)$ $-2$
$20$ $3$ $(1,2,3)$ $0$
$30$ $4$ $(1,2,3,4)$ $0$
$24$ $5$ $(1,2,3,4,5)$ $1$
$20$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.