Properties

Label 4.13941.6t13.b
Dimension $4$
Group $C_3^2:D_4$
Conductor $13941$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:\(13941\)\(\medspace = 3^{2} \cdot 1549 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.41823.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:D_4$
Parity: even
Projective image: $S_3\wr C_2$
Projective field: Galois closure of 6.0.41823.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 3 + 6\cdot 7 + 3\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 + 6\cdot 7 + 3\cdot 7^{2} + 2\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 a + \left(2 a + 2\right)\cdot 7 + \left(2 a + 5\right)\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + \left(4 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 a + 3 + \left(6 a + 3\right)\cdot 7 + \left(a + 5\right)\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(3 a + 6\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 5 }$ $=$ $ a + 6 + \left(4 a + 5\right)\cdot 7 + \left(4 a + 4\right)\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(2 a + 6\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 6 }$ $=$ $ a + 2 + 4\cdot 7 + 5 a\cdot 7^{2} + 4\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$

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

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

Character values on conjugacy classes

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