Properties

Label 2.675.12t11.a.b
Dimension $2$
Group $S_3 \times C_4$
Conductor $675$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3 \times C_4$
Conductor: \(675\)\(\medspace = 3^{3} \cdot 5^{2}\)
Artin stem field: 12.0.1037970703125.1
Galois orbit size: $2$
Smallest permutation container: $S_3 \times C_4$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.135.1

Defining polynomial

$f(x)$$=$\(x^{12} - 2 x^{9} + 4 x^{6} - 3 x^{3} + 1\)  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 $\mathbb{Q}_{ 13 }$: \(x^{4} + 3 x^{2} + 12 x + 2\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.