Properties

Label 12.118...757.18t218.a.a
Dimension $12$
Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor $1.181\times 10^{17}$
Root number $1$
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 stem field: Galois closure of 9.5.45487052759281.1
Galois orbit size: $1$
Smallest permutation container: 18T218
Parity: even
Determinant: 1.53.2t1.a.a
Projective image: $C_3^3.S_4$
Projective stem field: Galois closure of 9.5.45487052759281.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 7x^{7} - 7x^{6} - 7x^{5} + 84x^{4} + 35x^{3} - 280x^{2} + 28x + 280 \) Copy content Toggle raw display .

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 value
$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.