Properties

Label 4.13401.6t13.b
Dimension $4$
Group $C_3^2:D_4$
Conductor $13401$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:\(13401\)\(\medspace = 3^{2} \cdot 1489 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.40203.1
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 6.0.40203.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 8 a + 4 + \left(3 a + 8\right)\cdot 19 + 17\cdot 19^{2} + \left(13 a + 13\right)\cdot 19^{3} + \left(11 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 + 10\cdot 19 + 17\cdot 19^{2} + 15\cdot 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 + 13\cdot 19 + 14\cdot 19^{2} + 9\cdot 19^{3} + 9\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 12 + \left(15 a + 3\right)\cdot 19 + \left(18 a + 14\right)\cdot 19^{2} + \left(5 a + 7\right)\cdot 19^{3} + \left(7 a + 13\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 14 + \left(9 a + 18\right)\cdot 19 + \left(7 a + 6\right)\cdot 19^{2} + 16 a\cdot 19^{3} + \left(2 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 12 a + 2 + \left(9 a + 2\right)\cdot 19 + \left(11 a + 5\right)\cdot 19^{2} + \left(2 a + 9\right)\cdot 19^{3} + \left(16 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

Character values on conjugacy classes

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