Properties

Label 4.1777.5t5.a
Dimension $4$
Group $S_5$
Conductor $1777$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:\(1777\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.1777.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Projective image: $S_5$
Projective field: 5.1.1777.1

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 $\mathbb{Q}_{ 23 }$: \(x^{2} + 21 x + 5\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 3 a + 8 + \left(11 a + 8\right)\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(17 a + 13\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 20 a + 14 + \left(11 a + 4\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(9 a + 13\right)\cdot 23^{3} + \left(5 a + 11\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 3 + \left(16 a + 17\right)\cdot 23 + \left(12 a + 1\right)\cdot 23^{2} + \left(8 a + 3\right)\cdot 23^{3} + \left(18 a + 7\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 15 + 14\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 21 a + 7 + \left(6 a + 1\right)\cdot 23 + \left(10 a + 11\right)\cdot 23^{2} + \left(14 a + 7\right)\cdot 23^{3} + \left(4 a + 12\right)\cdot 23^{4} +O(23^{5})\)  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
$c1$
$1$ $1$ $()$ $4$
$10$ $2$ $(1,2)$ $2$
$15$ $2$ $(1,2)(3,4)$ $0$
$20$ $3$ $(1,2,3)$ $1$
$30$ $4$ $(1,2,3,4)$ $0$
$24$ $5$ $(1,2,3,4,5)$ $-1$
$20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.