Properties

Label 4.15028.6t13.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $15028$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:\(15028\)\(\medspace = 2^{2} \cdot 13 \cdot 17^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.255476.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:D_4$
Parity: even
Projective image: $\SOPlus(4,2)$
Projective field: Galois closure of 6.2.255476.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 15 a + 16 + \left(36 a + 2\right)\cdot 43 + \left(15 a + 28\right)\cdot 43^{2} + \left(37 a + 6\right)\cdot 43^{3} + \left(16 a + 27\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 28 a + 31 + \left(6 a + 23\right)\cdot 43 + \left(27 a + 7\right)\cdot 43^{2} + \left(5 a + 28\right)\cdot 43^{3} + \left(26 a + 6\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 30 + 41\cdot 43 + 28\cdot 43^{2} + 16\cdot 43^{3} + 34\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 12 + 33\cdot 43 + 13\cdot 43^{2} + 25\cdot 43^{3} + 31\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 30 a + 27 + \left(21 a + 39\right)\cdot 43 + \left(30 a + 20\right)\cdot 43^{2} + \left(28 a + 5\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 a + 14 + \left(21 a + 31\right)\cdot 43 + \left(12 a + 29\right)\cdot 43^{2} + \left(14 a + 3\right)\cdot 43^{3} + \left(36 a + 25\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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