Properties

Label 4.23616.6t13.b
Dimension $4$
Group $C_3^2:D_4$
Conductor $23616$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:\(23616\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 41 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.188928.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:D_4$
Parity: even
Projective image: $\SOPlus(4,2)$
Projective field: Galois closure of 6.2.188928.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 9 + 18\cdot 31 + 28\cdot 31^{2} + 20\cdot 31^{3} + 18\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 19 + 9\cdot 31 + 5\cdot 31^{2} + 5\cdot 31^{3} + 30\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a + 26 + \left(19 a + 18\right)\cdot 31 + \left(20 a + 5\right)\cdot 31^{2} + \left(12 a + 18\right)\cdot 31^{3} + \left(11 a + 16\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 a + 15 + \left(12 a + 17\right)\cdot 31 + \left(11 a + 7\right)\cdot 31^{2} + \left(a + 17\right)\cdot 31^{3} + \left(2 a + 14\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 30 a + 28 + \left(11 a + 24\right)\cdot 31 + \left(10 a + 27\right)\cdot 31^{2} + \left(18 a + 22\right)\cdot 31^{3} + \left(19 a + 26\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 24 a + 29 + \left(18 a + 3\right)\cdot 31 + \left(19 a + 18\right)\cdot 31^{2} + \left(29 a + 8\right)\cdot 31^{3} + \left(28 a + 17\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $4$
$6$ $2$ $(1,2)(3,4)(5,6)$ $0$
$6$ $2$ $(3,5)$ $2$
$9$ $2$ $(3,5)(4,6)$ $0$
$4$ $3$ $(1,3,5)(2,4,6)$ $-2$
$4$ $3$ $(1,3,5)$ $1$
$18$ $4$ $(1,2)(3,6,5,4)$ $0$
$12$ $6$ $(1,4,3,6,5,2)$ $0$
$12$ $6$ $(2,4,6)(3,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.