Properties

Label 2.29575.12t11.b.b
Dimension $2$
Group $S_3 \times C_4$
Conductor $29575$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3 \times C_4$
Conductor: \(29575\)\(\medspace = 5^{2} \cdot 7 \cdot 13^{2}\)
Artin stem field: 12.4.187441217958845703125.1
Galois orbit size: $2$
Smallest permutation container: $S_3 \times C_4$
Parity: odd
Determinant: 1.7.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.5915.1

Defining polynomial

$f(x)$$=$\(x^{12} - x^{11} - 7 x^{10} + 67 x^{9} - 143 x^{8} + 504 x^{7} + 1909 x^{6} + 4463 x^{5} + 5068 x^{4} - 8334 x^{3} - 5606 x^{2} + 2079 x + 1331\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \(x^{4} + 3 x^{2} + 19 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 9 a^{3} + 19 a^{2} + 7 a + 11 + \left(3 a^{3} + 9 a^{2} + 5 a + 1\right)\cdot 23 + \left(19 a^{3} + 14 a^{2} + 4 a + 15\right)\cdot 23^{2} + \left(15 a^{3} + 18 a^{2} + a + 8\right)\cdot 23^{3} + \left(6 a^{2} + a + 20\right)\cdot 23^{4} + \left(7 a^{3} + 4 a^{2} + 13 a + 3\right)\cdot 23^{5} + \left(9 a^{3} + 16 a^{2} + 9 a + 16\right)\cdot 23^{6} + \left(17 a^{3} + 7 a^{2} + 16 a + 15\right)\cdot 23^{7} + \left(19 a^{3} + 19 a^{2} + 4 a + 9\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 22 a^{3} + 19 a^{2} + 16 a + 12 + \left(5 a^{3} + 8 a^{2} + 8 a + 9\right)\cdot 23 + \left(21 a^{3} + 17 a^{2} + 13 a + 19\right)\cdot 23^{2} + \left(20 a^{2} + 22 a + 2\right)\cdot 23^{3} + \left(15 a^{3} + 3 a^{2} + 9 a + 15\right)\cdot 23^{4} + \left(14 a^{3} + 20 a^{2} + 2 a + 21\right)\cdot 23^{5} + \left(10 a^{3} + 14 a^{2} + 17 a + 12\right)\cdot 23^{6} + \left(10 a^{3} + 6 a^{2} + 14 a + 16\right)\cdot 23^{7} + \left(18 a^{3} + 12 a^{2} + 18 a + 18\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 12 a^{3} + 8 a^{2} + 7 a + 15 + \left(3 a^{3} + 8 a^{2} + 8 a + 11\right)\cdot 23 + \left(13 a^{3} + 7 a^{2} + 21 a + 5\right)\cdot 23^{2} + \left(10 a^{3} + 21 a^{2} + 21 a + 3\right)\cdot 23^{3} + \left(a^{3} + 16 a^{2} + 11 a + 15\right)\cdot 23^{4} + \left(9 a^{3} + 8 a + 7\right)\cdot 23^{5} + \left(a^{3} + 8 a + 5\right)\cdot 23^{6} + \left(11 a^{3} + 20 a^{2} + 22 a + 16\right)\cdot 23^{7} + \left(11 a^{3} + 17 a^{2} + 17 a\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 3 a^{3} + 8 a^{2} + 11 a + 18 + \left(2 a^{3} + 3 a^{2} + 2 a + 15\right)\cdot 23 + \left(9 a^{3} + 22 a^{2} + 12 a + 2\right)\cdot 23^{2} + \left(7 a^{3} + 13 a^{2} + 7 a + 10\right)\cdot 23^{3} + \left(18 a^{3} + 19 a^{2} + 7 a + 16\right)\cdot 23^{4} + \left(3 a^{3} + 3 a + 4\right)\cdot 23^{5} + \left(15 a^{3} + 4 a^{2} + 11 a + 15\right)\cdot 23^{6} + \left(19 a^{3} + 3 a^{2} + 5 a + 21\right)\cdot 23^{7} + \left(2 a^{3} + 6 a^{2} + 21 a + 22\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 16 a^{3} + 18 a^{2} + 11 a + \left(a^{3} + 6 a + 4\right)\cdot 23 + \left(5 a^{3} + 14 a^{2} + 18 a + 15\right)\cdot 23^{2} + \left(16 a^{3} + 2 a^{2} + 22 a + 1\right)\cdot 23^{3} + \left(17 a^{3} + 13 a^{2} + 5 a + 2\right)\cdot 23^{4} + \left(13 a^{3} + 10 a^{2} + 18 a\right)\cdot 23^{5} + \left(4 a^{3} + a^{2} + 14 a + 19\right)\cdot 23^{6} + \left(9 a^{3} + 2 a^{2} + 12 a + 16\right)\cdot 23^{7} + \left(17 a^{3} + 10 a^{2} + 9 a + 19\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 13 a^{3} + a^{2} + 20 a + 12 + \left(11 a^{3} + 14 a^{2} + 10 a + 22\right)\cdot 23 + \left(10 a^{3} + 16 a^{2} + 19 a + 2\right)\cdot 23^{2} + \left(18 a^{3} + 16 a^{2} + 14 a + 5\right)\cdot 23^{3} + \left(17 a^{3} + a^{2} + 4 a + 5\right)\cdot 23^{4} + \left(15 a^{3} + 6 a^{2} + 7 a + 22\right)\cdot 23^{5} + \left(12 a^{3} + 13 a^{2} + a + 10\right)\cdot 23^{6} + \left(20 a^{2} + 5\right)\cdot 23^{7} + \left(4 a^{3} + 16 a + 20\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 12 a^{3} + 12 a + 8 + \left(18 a^{3} + 15 a^{2} + 20 a + 6\right)\cdot 23 + \left(3 a^{3} + 15 a^{2} + a + 5\right)\cdot 23^{2} + \left(21 a^{3} + 16 a^{2} + 1\right)\cdot 23^{3} + \left(11 a^{3} + 11 a^{2} + 5 a + 21\right)\cdot 23^{4} + \left(a^{3} + 14 a^{2} + 7 a + 3\right)\cdot 23^{5} + \left(12 a^{3} + 20 a + 9\right)\cdot 23^{6} + \left(a^{3} + 14 a^{2} + 22 a + 17\right)\cdot 23^{7} + \left(5 a^{3} + 20 a^{2} + 2 a + 20\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 9 a^{3} + 9 a^{2} + 16 a + 19 + \left(22 a^{3} + 20 a^{2} + 13 a + 17\right)\cdot 23 + \left(17 a^{3} + a^{2} + 21 a + 19\right)\cdot 23^{2} + \left(15 a^{3} + 8 a^{2} + 21 a + 20\right)\cdot 23^{3} + \left(15 a^{3} + 14 a^{2} + 10 a + 3\right)\cdot 23^{4} + \left(16 a^{2} + 7 a + 18\right)\cdot 23^{5} + \left(20 a^{3} + 4 a^{2} + a + 13\right)\cdot 23^{6} + \left(17 a^{3} + 22 a^{2} + 17 a + 9\right)\cdot 23^{7} + \left(3 a^{3} + 18 a^{2} + 5 a + 11\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 21 a^{3} + 12 a^{2} + 21 a + 17 + \left(3 a^{3} + 10 a^{2} + 18 a + 14\right)\cdot 23 + \left(2 a^{3} + 15 a^{2} + 14 a + 10\right)\cdot 23^{2} + \left(9 a^{3} + 4 a^{2} + 22 a + 20\right)\cdot 23^{3} + \left(16 a^{3} + 7 a^{2} + 17 a + 17\right)\cdot 23^{4} + \left(19 a^{3} + 3 a^{2} + 22 a + 13\right)\cdot 23^{5} + \left(19 a^{3} + 22 a^{2} + 13 a + 2\right)\cdot 23^{6} + \left(2 a^{2} + 21 a + 12\right)\cdot 23^{7} + \left(10 a^{3} + 20 a^{2} + 7 a + 6\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 21 a^{3} + 12 a^{2} + 13 a + 17 + \left(14 a^{3} + 8 a^{2} + 3 a + 1\right)\cdot 23 + \left(4 a^{2} + 1\right)\cdot 23^{2} + \left(20 a^{3} + a^{2} + 7 a + 10\right)\cdot 23^{3} + \left(22 a^{3} + 2 a^{2} + 18 a + 10\right)\cdot 23^{4} + \left(13 a^{3} + 13 a^{2} + 10 a + 4\right)\cdot 23^{5} + \left(13 a^{3} + 4 a^{2} + 15 a + 2\right)\cdot 23^{6} + \left(22 a^{3} + 7 a^{2} + 6 a + 6\right)\cdot 23^{7} + \left(12 a^{3} + 6 a^{2} + a + 16\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 11 }$ $=$ \( 15 a^{3} + 14 a^{2} + 5 a + 15 + \left(18 a^{2} + 15 a + 20\right)\cdot 23 + \left(7 a^{3} + 18 a^{2} + 9 a + 15\right)\cdot 23^{2} + \left(6 a^{3} + 18 a^{2} + 17 a + 7\right)\cdot 23^{3} + \left(5 a^{3} + 19 a^{2} + 20 a + 16\right)\cdot 23^{4} + \left(3 a^{3} + 5 a^{2} + 3 a + 12\right)\cdot 23^{5} + \left(11 a^{3} + 19 a^{2} + 8 a\right)\cdot 23^{6} + \left(11 a^{3} + 15 a^{2} + 18 a + 16\right)\cdot 23^{7} + \left(11 a^{3} + a^{2} + 18 a + 22\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display
$r_{ 12 }$ $=$ \( 8 a^{3} + 18 a^{2} + 22 a + 18 + \left(3 a^{3} + 19 a^{2} + 11\right)\cdot 23 + \left(5 a^{3} + 12 a^{2} + a + 1\right)\cdot 23^{2} + \left(19 a^{3} + 17 a^{2} + a\right)\cdot 23^{3} + \left(17 a^{3} + 20 a^{2} + a + 17\right)\cdot 23^{4} + \left(11 a^{3} + 18 a^{2} + 10 a + 1\right)\cdot 23^{5} + \left(7 a^{3} + 13 a^{2} + 16 a + 7\right)\cdot 23^{6} + \left(15 a^{3} + 15 a^{2} + 2 a + 7\right)\cdot 23^{7} + \left(20 a^{3} + 3 a^{2} + 13 a + 14\right)\cdot 23^{8} +O(23^{9})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.