Properties

Label 2.1260.12t18.e
Dimension $2$
Group $C_6\times S_3$
Conductor $1260$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$C_6\times S_3$
Conductor:\(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Artin number field: Galois closure of 12.0.3087580356000000.2
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.11340.2

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:
$r_{ 1 }$ $=$ \( 12 a^{5} + 2 a^{2} + 6 a + 6 + \left(3 a^{5} + 9 a^{4} + a^{3} + 2 a^{2} + 11 a + 9\right)\cdot 13 + \left(2 a^{5} + 10 a^{4} + 7 a^{3} + 7 a^{2} + 8 a + 2\right)\cdot 13^{2} + \left(5 a^{5} + 5 a^{4} + a^{3} + 6 a^{2} + 5 a + 11\right)\cdot 13^{3} + \left(8 a^{5} + 9 a^{4} + 10 a^{3} + 11 a^{2} + a + 1\right)\cdot 13^{4} + \left(8 a^{4} + 12 a^{3} + 7 a^{2} + 4 a + 8\right)\cdot 13^{5} + \left(9 a^{5} + 7 a^{4} + 5 a^{3} + 2 a^{2} + 5 a + 6\right)\cdot 13^{6} + \left(2 a^{5} + 4 a^{4} + 8 a^{3} + 3 a^{2} + 3 a + 7\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a^{5} + 10 a^{4} + 9 a^{3} + 4 a^{2} + 3 a + 8 + \left(12 a^{5} + 11 a^{4} + 6 a^{3} + 6 a^{2} + 3 a + 8\right)\cdot 13 + \left(7 a^{5} + 11 a^{4} + 10 a^{3} + 11 a^{2} + 12 a\right)\cdot 13^{2} + \left(2 a^{5} + 8 a^{4} + a^{3} + 9 a^{2} + 11 a + 3\right)\cdot 13^{3} + \left(11 a^{5} + 12 a^{4} + 3 a^{3} + 7 a^{2} + 11 a + 1\right)\cdot 13^{4} + \left(a^{5} + 2 a^{4} + 2 a^{3} + 11 a^{2} + 10 a + 11\right)\cdot 13^{5} + \left(9 a^{5} + 7 a^{4} + 12 a^{3} + 5 a^{2} + 10 a + 4\right)\cdot 13^{6} + \left(9 a^{5} + 5 a^{4} + a^{3} + 6 a^{2} + 10 a + 9\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a^{5} + 5 a^{4} + 5 a^{3} + 3 a + 12 + \left(3 a^{4} + 2 a^{3} + 12 a^{2} + 3 a + 1\right)\cdot 13 + \left(9 a^{5} + 7 a^{3} + 6 a^{2} + a + 8\right)\cdot 13^{2} + \left(9 a^{4} + 11 a^{3} + 6 a^{2} + 10 a + 4\right)\cdot 13^{3} + \left(3 a^{5} + 4 a^{4} + 10 a^{3} + a^{2} + 11 a + 6\right)\cdot 13^{4} + \left(2 a^{5} + 8 a^{4} + 4 a^{3} + 8 a^{2} + 8\right)\cdot 13^{5} + \left(10 a^{4} + 5 a^{3} + a^{2} + 6 a + 4\right)\cdot 13^{6} + \left(4 a^{5} + 4 a^{4} + 12 a^{3} + 2 a^{2} + 7 a + 6\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a^{5} + 3 a^{4} + 8 a^{3} + 3 a^{2} + 8 a + 4 + \left(6 a^{5} + 5 a^{4} + 5 a^{3} + 9 a + 7\right)\cdot 13 + \left(10 a^{5} + 6 a^{4} + 11 a^{3} + 2 a^{2} + a + 11\right)\cdot 13^{2} + \left(3 a^{5} + 7 a^{4} + 5 a^{3} + 4 a^{2} + 6 a + 10\right)\cdot 13^{3} + \left(5 a^{5} + 11 a^{4} + 5 a^{3} + 4 a^{2} + 8 a + 6\right)\cdot 13^{4} + \left(3 a^{5} + 6 a^{4} + 12 a^{3} + 9 a + 9\right)\cdot 13^{5} + \left(9 a^{5} + 3 a^{4} + 8 a^{3} + 7 a^{2} + 9 a + 1\right)\cdot 13^{6} + \left(2 a^{5} + 2 a^{4} + 7 a^{2} + a + 8\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a^{5} + 6 a^{4} + 3 a^{3} + 4 a^{2} + 10 a + 2 + \left(12 a^{5} + 8 a^{4} + 7 a^{3} + 6 a^{2} + 7 a\right)\cdot 13 + \left(2 a^{5} + 9 a^{4} + 8 a^{3} + 11 a^{2} + 5 a + 4\right)\cdot 13^{2} + \left(a^{5} + 12 a^{4} + 10 a^{3} + 11 a^{2} + 5 a + 6\right)\cdot 13^{3} + \left(7 a^{5} + 10 a^{4} + a^{2} + 4\right)\cdot 13^{4} + \left(4 a^{5} + 3 a^{4} + 11 a^{3} + 7 a^{2} + 9 a + 10\right)\cdot 13^{5} + \left(10 a^{5} + 11 a^{3} + 12 a^{2} + 8 a + 6\right)\cdot 13^{6} + \left(10 a^{4} + 4 a^{3} + 12 a^{2} + 10 a + 3\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 12 a^{5} + 4 a^{4} + 2 a^{3} + 3 a + 2 + \left(6 a^{5} + 6 a^{4} + a^{3} + 6 a + 7\right)\cdot 13 + \left(3 a^{5} + 5 a^{3} + 5 a^{2} + 8 a + 11\right)\cdot 13^{2} + \left(8 a^{5} + 12 a^{4} + 3 a^{3} + 11 a^{2} + 10 a + 9\right)\cdot 13^{3} + \left(5 a^{5} + 8 a^{4} + 8 a^{3} + 8 a^{2} + 7 a + 6\right)\cdot 13^{4} + \left(3 a^{5} + 11 a^{4} + 8 a^{2} + 4 a + 5\right)\cdot 13^{5} + \left(12 a^{5} + 3 a^{4} + a^{2} + a\right)\cdot 13^{6} + \left(3 a^{5} + 12 a^{4} + 3 a^{3} + 3 a^{2} + 11 a + 7\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 9 a^{5} + a^{4} + 7 a^{3} + 2 a^{2} + 3 a + 10 + \left(3 a^{5} + 6 a^{4} + 12 a^{3} + 10 a^{2} + 2 a + 6\right)\cdot 13 + \left(2 a^{5} + 2 a^{4} + 10 a^{3} + 12 a^{2} + 9 a + 11\right)\cdot 13^{2} + \left(2 a^{5} + 5 a^{4} + 12 a^{3} + 5 a^{2} + 3 a\right)\cdot 13^{3} + \left(9 a^{5} + 7 a^{4} + a + 11\right)\cdot 13^{4} + \left(7 a^{5} + 4 a^{4} + a^{3} + 12 a^{2} + 4 a + 9\right)\cdot 13^{5} + \left(4 a^{5} + 6 a^{4} + 4 a^{3} + 7 a^{2} + 7 a\right)\cdot 13^{6} + \left(4 a^{5} + 5 a^{4} + a^{3} + 6 a^{2} + 12 a\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 10 a^{4} + 5 a^{3} + 8 a^{2} + 12 a + 3 + \left(3 a^{5} + 11 a^{4} + 6 a^{3} + 10 a^{2} + 4 a + 9\right)\cdot 13 + \left(8 a^{4} + 7 a^{3} + 3 a^{2} + 2 a + 11\right)\cdot 13^{2} + \left(4 a^{5} + 12 a^{4} + 5 a^{3} + 2 a^{2} + a + 3\right)\cdot 13^{3} + \left(12 a^{5} + 4 a^{4} + 10 a^{3} + 10 a^{2} + 3 a + 4\right)\cdot 13^{4} + \left(8 a^{5} + 10 a^{4} + 4 a^{2} + 12 a + 8\right)\cdot 13^{5} + \left(7 a^{5} + a^{4} + 11 a^{3} + 3 a^{2} + 10 a + 4\right)\cdot 13^{6} + \left(7 a^{5} + 6 a^{4} + 3 a^{3} + 2 a^{2} + 7 a + 10\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 11 a^{5} + 3 a^{4} + 8 a^{3} + 9 a^{2} + 9 + \left(6 a^{5} + 11 a^{4} + 4 a^{3} + 6 a^{2} + 12 a + 5\right)\cdot 13 + \left(6 a^{5} + 2 a^{4} + 12 a^{3} + 9 a^{2} + 11 a + 10\right)\cdot 13^{2} + \left(3 a^{5} + a^{4} + 11 a^{3} + 2 a^{2} + 9 a + 9\right)\cdot 13^{3} + \left(6 a^{4} + 3 a^{3} + 2 a^{2} + 4 a + 1\right)\cdot 13^{4} + \left(5 a^{5} + 10 a^{4} + a^{3} + 10 a^{2} + 12 a + 10\right)\cdot 13^{5} + \left(3 a^{5} + 8 a^{4} + a^{3} + 11 a^{2} + 2 a + 5\right)\cdot 13^{6} + \left(8 a^{5} + 3 a^{4} + 5 a^{3} + 9 a^{2} + 4 a + 2\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 8 a^{5} + 2 a^{4} + 10 a^{3} + 7 a^{2} + 7 a + 8 + \left(9 a^{5} + 8 a^{4} + 6 a^{3} + 9 a^{2} + 7 a + 10\right)\cdot 13 + \left(2 a^{5} + 11 a^{4} + 4 a^{3} + a^{2} + 4 a\right)\cdot 13^{2} + \left(8 a^{5} + 11 a^{4} + 11 a^{3} + 10 a^{2} + 10 a + 9\right)\cdot 13^{3} + \left(5 a^{5} + 5 a^{4} + 8 a^{3} + 4 a^{2} + 12 a\right)\cdot 13^{4} + \left(3 a^{5} + 5 a^{4} + 9 a^{3} + 2 a^{2} + 10 a + 5\right)\cdot 13^{5} + \left(12 a^{5} + 12 a^{4} + 9 a^{3} + 12 a^{2} + 7 a + 7\right)\cdot 13^{6} + \left(11 a^{5} + a^{4} + 9 a^{3} + 12 a^{2} + 2 a + 3\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 7 a^{5} + 3 a^{4} + 12 a^{2} + 11 a + 6 + \left(10 a^{5} + 4 a^{3} + 8 a^{2} + 6 a + 3\right)\cdot 13 + \left(7 a^{5} + 7 a^{4} + 8 a^{3} + 8 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(10 a^{5} + 12 a^{3} + 8 a^{2} + 9 a + 8\right)\cdot 13^{3} + \left(10 a^{5} + 8 a^{4} + 8 a + 2\right)\cdot 13^{4} + \left(6 a^{5} + 8 a^{4} + a^{3} + 10 a^{2} + 5 a + 1\right)\cdot 13^{5} + \left(5 a^{5} + 10 a^{4} + 8 a^{3} + a + 11\right)\cdot 13^{6} + \left(12 a^{5} + 11 a^{4} + 10 a^{3} + 10 a^{2} + 12 a + 8\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 10 a^{5} + 5 a^{4} + 8 a^{3} + a^{2} + 12 a + 8 + \left(a^{5} + 9 a^{4} + 6 a^{3} + 5 a^{2} + 2 a + 7\right)\cdot 13 + \left(9 a^{5} + 5 a^{4} + 10 a^{3} + 10 a^{2} + 3 a\right)\cdot 13^{2} + \left(a^{5} + 3 a^{4} + a^{3} + 10 a^{2} + 6 a\right)\cdot 13^{3} + \left(12 a^{5} + a^{3} + 10 a^{2} + 5 a + 4\right)\cdot 13^{4} + \left(3 a^{5} + 9 a^{4} + 7 a^{3} + 7 a^{2} + 6 a + 3\right)\cdot 13^{5} + \left(7 a^{5} + 4 a^{4} + 12 a^{3} + 10 a^{2} + 5 a + 10\right)\cdot 13^{6} + \left(9 a^{5} + 9 a^{4} + 2 a^{3} + 6 a + 10\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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