Properties

Label 4.1516563.12t34.b
Dimension $4$
Group $C_3^2:D_4$
Conductor $1516563$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:\(1516563\)\(\medspace = 3^{5} \cdot 79^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.57591.1
Galois orbit size: $1$
Smallest permutation container: 12T34
Parity: odd
Projective image: $\SOPlus(4,2)$
Projective field: Galois closure of 6.0.57591.1

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:
$r_{ 1 }$ $=$ \( 27 + 20\cdot 31 + 29\cdot 31^{2} + 10\cdot 31^{3} + 16\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a + 9 + \left(28 a + 19\right)\cdot 31 + \left(12 a + 17\right)\cdot 31^{2} + \left(6 a + 25\right)\cdot 31^{3} + \left(16 a + 9\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 + 31 + 8\cdot 31^{2} + 10\cdot 31^{3} + 19\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 17 + \left(8 a + 9\right)\cdot 31 + 15 a\cdot 31^{2} + \left(28 a + 5\right)\cdot 31^{3} + \left(5 a + 14\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 7 a + 26 + \left(2 a + 21\right)\cdot 31 + \left(18 a + 14\right)\cdot 31^{2} + \left(24 a + 25\right)\cdot 31^{3} + \left(14 a + 4\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 25 a + 29 + \left(22 a + 19\right)\cdot 31 + \left(15 a + 22\right)\cdot 31^{2} + \left(2 a + 15\right)\cdot 31^{3} + \left(25 a + 28\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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