Properties

Label 12.118...757.18t218.a
Dimension $12$
Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor $1.181\times 10^{17}$
Indicator $1$

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Basic invariants

Dimension:$12$
Group:$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor:\(118129876015852757\)\(\medspace = 7^{10} \cdot 53^{5}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 9.5.45487052759281.1
Galois orbit size: $1$
Smallest permutation container: 18T218
Parity: even
Projective image: $C_3^3.S_4$
Projective field: Galois closure of 9.5.45487052759281.1

Galois action

Roots of defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character values
$c1$
$1$ $1$ $()$ $12$
$27$ $2$ $(1,6)(4,5)$ $0$
$54$ $2$ $(1,6)(2,4)(3,5)(8,9)$ $2$
$6$ $3$ $(1,7,6)$ $0$
$8$ $3$ $(1,6,7)(2,3,9)(4,5,8)$ $3$
$12$ $3$ $(1,7,6)(4,8,5)$ $-3$
$72$ $3$ $(1,2,4)(3,5,6)(7,9,8)$ $0$
$54$ $4$ $(1,5,6,4)(7,8)$ $0$
$54$ $6$ $(1,6,7)(2,3)(4,5)$ $0$
$108$ $6$ $(1,8,7,5,6,4)(2,3)$ $-1$
$72$ $9$ $(1,3,5,6,9,8,7,2,4)$ $0$
$72$ $9$ $(1,9,8,7,3,5,6,2,4)$ $0$
$54$ $12$ $(1,7,6)(2,5,3,4)(8,9)$ $0$
$54$ $12$ $(1,6,7)(2,5,3,4)(8,9)$ $0$
The blue line marks the conjugacy class containing complex conjugation.