Properties

Label 2.3040.12t18.c
Dimension $2$
Group $C_6\times S_3$
Conductor $3040$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$C_6\times S_3$
Conductor:\(3040\)\(\medspace = 2^{5} \cdot 5 \cdot 19 \)
Artin number field: Galois closure of 12.0.34162868224000000.1
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.14440.1

Galois action

Roots of defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

Cycle notation
$(1,7)(2,11)(3,12)(4,8)(5,9)(6,10)$
$(1,6,3,4,5,2)(7,10,12,8,9,11)$
$(2,4,6)(8,10,11)$
$(1,3,5)(2,6,4)(7,12,9)(8,11,10)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 12 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,7)(2,11)(3,12)(4,8)(5,9)(6,10)$ $-2$ $-2$
$3$ $2$ $(1,4)(2,3)(5,6)(7,8)(9,10)(11,12)$ $0$ $0$
$3$ $2$ $(1,8)(2,12)(3,11)(4,7)(5,10)(6,9)$ $0$ $0$
$1$ $3$ $(1,3,5)(2,6,4)(7,12,9)(8,11,10)$ $2 \zeta_{3}$ $-2 \zeta_{3} - 2$
$1$ $3$ $(1,5,3)(2,4,6)(7,9,12)(8,10,11)$ $-2 \zeta_{3} - 2$ $2 \zeta_{3}$
$2$ $3$ $(2,4,6)(8,10,11)$ $-\zeta_{3}$ $\zeta_{3} + 1$
$2$ $3$ $(2,6,4)(8,11,10)$ $\zeta_{3} + 1$ $-\zeta_{3}$
$2$ $3$ $(1,5,3)(2,6,4)(7,9,12)(8,11,10)$ $-1$ $-1$
$1$ $6$ $(1,12,5,7,3,9)(2,10,4,11,6,8)$ $-2 \zeta_{3}$ $2 \zeta_{3} + 2$
$1$ $6$ $(1,9,3,7,5,12)(2,8,6,11,4,10)$ $2 \zeta_{3} + 2$ $-2 \zeta_{3}$
$2$ $6$ $(1,7)(2,8,6,11,4,10)(3,12)(5,9)$ $\zeta_{3}$ $-\zeta_{3} - 1$
$2$ $6$ $(1,7)(2,10,4,11,6,8)(3,12)(5,9)$ $-\zeta_{3} - 1$ $\zeta_{3}$
$2$ $6$ $(1,9,3,7,5,12)(2,10,4,11,6,8)$ $1$ $1$
$3$ $6$ $(1,6,3,4,5,2)(7,10,12,8,9,11)$ $0$ $0$
$3$ $6$ $(1,2,5,4,3,6)(7,11,9,8,12,10)$ $0$ $0$
$3$ $6$ $(1,10,3,8,5,11)(2,7,6,12,4,9)$ $0$ $0$
$3$ $6$ $(1,11,5,8,3,10)(2,9,4,12,6,7)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.