Properties

Label 4.363888.6t13.b
Dimension $4$
Group $C_3^2:D_4$
Conductor $363888$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:\(363888\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.20741616.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:D_4$
Parity: even
Projective image: $\SOPlus(4,2)$
Projective field: Galois closure of 6.2.20741616.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: \( x^{2} + 101x + 3 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 88 a + 34 + \left(76 a + 104\right)\cdot 113 + \left(76 a + 100\right)\cdot 113^{2} + \left(64 a + 102\right)\cdot 113^{3} + \left(54 a + 28\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 36 a + 106 + \left(100 a + 23\right)\cdot 113 + \left(17 a + 105\right)\cdot 113^{2} + \left(45 a + 5\right)\cdot 113^{3} + \left(111 a + 57\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 104 + 98\cdot 113 + 55\cdot 113^{2} + 98\cdot 113^{3} + 76\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 50 + 16\cdot 113 + 43\cdot 113^{2} + 42\cdot 113^{3} + 16\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 77 a + 86 + \left(12 a + 61\right)\cdot 113 + \left(95 a + 106\right)\cdot 113^{2} + \left(67 a + 77\right)\cdot 113^{3} + \left(a + 105\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 25 a + 73 + \left(36 a + 33\right)\cdot 113 + \left(36 a + 40\right)\cdot 113^{2} + \left(48 a + 11\right)\cdot 113^{3} + \left(58 a + 54\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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