Properties

Label 5.31956409.10t13.a
Dimension $5$
Group $S_5$
Conductor $31956409$
Indicator $1$

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Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:\(31956409\)\(\medspace = 5653^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.5653.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Projective image: $S_5$
Projective field: Galois closure of 5.1.5653.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 21 a + 20 + \left(10 a + 10\right)\cdot 23 + \left(4 a + 15\right)\cdot 23^{2} + \left(18 a + 17\right)\cdot 23^{3} + 8 a\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 2\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 16 + \left(12 a + 11\right)\cdot 23 + \left(18 a + 13\right)\cdot 23^{2} + \left(4 a + 3\right)\cdot 23^{3} + 14 a\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 a + 9 + \left(21 a + 17\right)\cdot 23 + \left(5 a + 14\right)\cdot 23^{2} + \left(19 a + 21\right)\cdot 23^{3} + \left(17 a + 14\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 12 a + 8 + \left(a + 3\right)\cdot 23 + \left(17 a + 5\right)\cdot 23^{2} + \left(3 a + 8\right)\cdot 23^{3} + \left(5 a + 8\right)\cdot 23^{4} +O(23^{5})\) Copy content 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
$c1$
$1$ $1$ $()$ $5$
$10$ $2$ $(1,2)$ $1$
$15$ $2$ $(1,2)(3,4)$ $1$
$20$ $3$ $(1,2,3)$ $-1$
$30$ $4$ $(1,2,3,4)$ $-1$
$24$ $5$ $(1,2,3,4,5)$ $0$
$20$ $6$ $(1,2,3)(4,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.