Properties

Label 2.405.12t18.a.a
Dimension $2$
Group $C_6\times S_3$
Conductor $405$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(405\)\(\medspace = 3^{4} \cdot 5 \)
Artin stem field: 12.0.6053445140625.2
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.45.6t1.b.b
Projective image: $S_3$
Projective stem field: 3.1.135.1

Defining polynomial

$f(x)$$=$\(x^{12} - x^{9} + 2 x^{6} + x^{3} + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2\)  Toggle raw display

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 12 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)(9,12)(10,11)$$-2$
$3$$2$$(1,10)(2,9)(3,4)(5,6)(7,12)(8,11)$$0$
$3$$2$$(1,11)(2,12)(3,5)(4,6)(7,9)(8,10)$$0$
$1$$3$$(1,9,5)(2,6,10)(3,11,7)(4,8,12)$$-2 \zeta_{3} - 2$
$1$$3$$(1,5,9)(2,10,6)(3,7,11)(4,12,8)$$2 \zeta_{3}$
$2$$3$$(1,9,5)(2,10,6)(3,7,11)(4,8,12)$$-1$
$2$$3$$(2,6,10)(3,11,7)$$-\zeta_{3}$
$2$$3$$(2,10,6)(3,7,11)$$\zeta_{3} + 1$
$1$$6$$(1,4,9,8,5,12)(2,11,6,7,10,3)$$-2 \zeta_{3}$
$1$$6$$(1,12,5,8,9,4)(2,3,10,7,6,11)$$2 \zeta_{3} + 2$
$2$$6$$(1,12,5,8,9,4)(2,11,6,7,10,3)$$1$
$2$$6$$(1,8)(2,3,10,7,6,11)(4,5)(9,12)$$\zeta_{3}$
$2$$6$$(1,8)(2,11,6,7,10,3)(4,5)(9,12)$$-\zeta_{3} - 1$
$3$$6$$(1,6,9,10,5,2)(3,12,11,4,7,8)$$0$
$3$$6$$(1,2,5,10,9,6)(3,8,7,4,11,12)$$0$
$3$$6$$(1,3,9,11,5,7)(2,8,6,12,10,4)$$0$
$3$$6$$(1,7,5,11,9,3)(2,4,10,12,6,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.