Properties

Label 5.107495424.6t15.a
Dimension $5$
Group $A_6$
Conductor $107495424$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:\(107495424\)\(\medspace = 2^{14} \cdot 3^{8} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.107495424.3
Galois orbit size: $1$
Smallest permutation container: $A_6$
Parity: even
Projective image: $A_6$
Projective field: Galois closure of 6.2.107495424.3

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 4 a + 30 + \left(18 a + 31\right)\cdot 47 + \left(2 a + 30\right)\cdot 47^{2} + \left(14 a + 4\right)\cdot 47^{3} + \left(6 a + 22\right)\cdot 47^{4} + \left(34 a + 16\right)\cdot 47^{5} +O(47^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 a + 35 + \left(16 a + 43\right)\cdot 47 + \left(43 a + 42\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(18 a + 16\right)\cdot 47^{4} + \left(a + 21\right)\cdot 47^{5} +O(47^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 + 12\cdot 47 + 22\cdot 47^{2} + 20\cdot 47^{3} + 13\cdot 47^{4} + 47^{5} +O(47^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 24 + 12\cdot 47 + 8\cdot 47^{2} + 44\cdot 47^{4} + 17\cdot 47^{5} +O(47^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 42 a + 45 + \left(30 a + 23\right)\cdot 47 + \left(3 a + 19\right)\cdot 47^{2} + \left(18 a + 26\right)\cdot 47^{3} + \left(28 a + 24\right)\cdot 47^{4} + \left(45 a + 5\right)\cdot 47^{5} +O(47^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 43 a + 38 + \left(28 a + 16\right)\cdot 47 + \left(44 a + 17\right)\cdot 47^{2} + \left(32 a + 30\right)\cdot 47^{3} + \left(40 a + 20\right)\cdot 47^{4} + \left(12 a + 31\right)\cdot 47^{5} +O(47^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $5$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$
$40$ $3$ $(1,2,3)$ $2$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$72$ $5$ $(1,2,3,4,5)$ $0$
$72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.