Properties

Label 3.107653.12t29.a.a
Dimension $3$
Group $C_4\times A_4$
Conductor $107653$
Root number not computed
Indicator $0$

Related objects

Learn more about

Basic invariants

Dimension: $3$
Group: 12T29
Conductor: \(107653\)\(\medspace = 7^{2} \cdot 13^{3}\)
Artin stem field: 12.8.61132828589969773.1
Galois orbit size: $2$
Smallest permutation container: 12T29
Parity: odd
Determinant: 1.13.4t1.a.a
Projective image: $A_4$
Projective stem field: 4.0.8281.1

Defining polynomial

$f(x)$$=$\(x^{12} - 3 x^{11} - 9 x^{10} + 29 x^{9} + 41 x^{8} - 82 x^{7} - 263 x^{6} + 36 x^{5} + 705 x^{4} + 468 x^{3} - 182 x^{2} - 169 x - 13\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \(x^{4} + 23 x + 6\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 23 a^{3} + 24 a^{2} + 20 a + 3 + \left(36 a^{3} + 36 a^{2} + 13 a + 25\right)\cdot 41 + \left(38 a^{3} + 17 a^{2} + 40 a + 29\right)\cdot 41^{2} + \left(37 a^{3} + 32 a^{2} + 7 a + 35\right)\cdot 41^{3} + \left(24 a^{3} + 3 a^{2} + 32 a + 4\right)\cdot 41^{4} + \left(9 a^{3} + 27 a^{2} + 16 a + 18\right)\cdot 41^{5} + \left(7 a^{3} + 23 a^{2} + 18 a + 8\right)\cdot 41^{6} + \left(4 a^{3} + 32 a^{2} + 15 a + 5\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 7 a^{3} + 13 a^{2} + 6 a + 27 + \left(a^{3} + 37 a^{2} + 13 a + 19\right)\cdot 41 + \left(27 a^{3} + 17 a^{2} + 29 a + 39\right)\cdot 41^{2} + \left(15 a^{3} + 32 a^{2} + 16 a + 21\right)\cdot 41^{3} + \left(19 a^{3} + 3 a^{2} + 24 a + 24\right)\cdot 41^{4} + \left(26 a^{3} + 5 a^{2} + a + 12\right)\cdot 41^{5} + \left(a^{3} + 26 a^{2} + 11 a + 17\right)\cdot 41^{6} + \left(26 a^{3} + 13 a^{2} + 22 a + 13\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 a^{3} + 39 a^{2} + 28 a + 29 + \left(13 a^{3} + 22 a^{2} + 39 a + 24\right)\cdot 41 + \left(5 a^{3} + 7 a^{2} + 11 a + 24\right)\cdot 41^{2} + \left(12 a^{3} + 24 a^{2} + 18 a + 10\right)\cdot 41^{3} + \left(25 a^{3} + 29 a^{2} + 27 a + 11\right)\cdot 41^{4} + \left(20 a^{3} + 13 a^{2} + 40 a + 13\right)\cdot 41^{5} + \left(37 a^{3} + 2 a^{2} + 27 a + 28\right)\cdot 41^{6} + \left(16 a^{3} + 4 a^{2} + 9\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 39 a^{3} + 20 a^{2} + 20 a + 13 + \left(39 a^{3} + 31 a^{2} + 5 a + 24\right)\cdot 41 + \left(22 a^{3} + 11 a^{2} + 40 a + 39\right)\cdot 41^{2} + \left(7 a^{3} + 27 a + 33\right)\cdot 41^{3} + \left(36 a^{3} + 34 a^{2} + 38 a + 36\right)\cdot 41^{4} + \left(33 a^{3} + 33 a^{2} + 20 a + 17\right)\cdot 41^{5} + \left(13 a^{3} + 24 a^{2} + 11 a + 17\right)\cdot 41^{6} + \left(24 a^{3} + 6 a^{2} + 22 a + 20\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 21 a^{3} + 2 a^{2} + 20 a + 10 + \left(20 a^{3} + 7 a^{2} + 8 a + 27\right)\cdot 41 + \left(6 a^{3} + 29 a^{2} + 18 a + 11\right)\cdot 41^{2} + \left(24 a^{3} + 9 a^{2} + 17 a + 33\right)\cdot 41^{3} + \left(23 a^{2} + 2 a + 37\right)\cdot 41^{4} + \left(23 a^{3} + 18 a^{2} + 23 a + 14\right)\cdot 41^{5} + \left(25 a^{3} + 15 a^{2} + 32 a + 35\right)\cdot 41^{6} + \left(5 a^{3} + 37 a^{2} + 39 a + 35\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 4 a^{3} + 39 a^{2} + 6 a + 6 + \left(39 a^{3} + 6 a^{2} + 16 a + 28\right)\cdot 41 + \left(4 a^{3} + 27 a^{2} + 22 a + 27\right)\cdot 41^{2} + \left(11 a^{3} + 4 a^{2} + 5 a + 25\right)\cdot 41^{3} + \left(a^{3} + 7 a^{2} + 8 a + 19\right)\cdot 41^{4} + \left(31 a^{3} + 40 a^{2} + 13 a + 19\right)\cdot 41^{5} + \left(40 a^{3} + 6 a^{2} + 9 a + 15\right)\cdot 41^{6} + \left(32 a^{3} + 8 a^{2} + 3 a + 10\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 33 a^{3} + 13 a^{2} + 19 a + 32 + \left(37 a^{3} + 38 a^{2} + 24 a + 15\right)\cdot 41 + \left(16 a^{3} + 12 a^{2} + 21 a + 9\right)\cdot 41^{2} + \left(30 a^{3} + 11 a^{2} + 32 a + 8\right)\cdot 41^{3} + \left(29 a^{3} + 6 a^{2} + 38 a + 16\right)\cdot 41^{4} + \left(10 a^{3} + 8 a^{2} + a + 37\right)\cdot 41^{5} + \left(27 a^{3} + 11 a^{2} + 10 a + 25\right)\cdot 41^{6} + \left(30 a^{3} + 8 a^{2} + 30 a + 31\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 27 a^{3} + 8 a^{2} + 25 a + 3 + \left(6 a^{3} + 35 a^{2} + 25 a + 22\right)\cdot 41 + \left(2 a^{3} + 14 a^{2} + 21 a + 20\right)\cdot 41^{2} + \left(32 a^{3} + 30 a^{2} + 13 a + 7\right)\cdot 41^{3} + \left(20 a^{3} + 27 a^{2} + 22 a + 28\right)\cdot 41^{4} + \left(35 a^{3} + 24 a^{2} + 2 a + 14\right)\cdot 41^{5} + \left(23 a^{3} + 18 a^{2} + 29 a + 21\right)\cdot 41^{6} + \left(30 a^{3} + 40 a^{2} + a + 40\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 11 a^{3} + 6 a^{2} + 15 a + 1 + \left(35 a^{3} + 25 a^{2} + 4 a + 13\right)\cdot 41 + \left(20 a^{3} + 2 a^{2} + 8 a + 36\right)\cdot 41^{2} + \left(a^{3} + 14 a^{2} + 23 a + 11\right)\cdot 41^{3} + \left(2 a^{3} + a^{2} + 24 a + 30\right)\cdot 41^{4} + \left(33 a^{2} + 22 a + 37\right)\cdot 41^{5} + \left(10 a^{3} + 3 a^{2} + 25 a + 4\right)\cdot 41^{6} + \left(30 a^{3} + 37 a^{2} + 35 a + 14\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 4 a^{3} + 15 a^{2} + 18 a + 14 + \left(19 a^{3} + 33 a^{2} + 17 a + 23\right)\cdot 41 + \left(29 a^{3} + 7 a^{2} + 27 a + 28\right)\cdot 41^{2} + \left(34 a^{3} + 37 a^{2} + 18 a + 20\right)\cdot 41^{3} + \left(12 a^{3} + 18 a^{2} + 20 a + 13\right)\cdot 41^{4} + \left(27 a^{3} + 5 a^{2} + 17 a + 17\right)\cdot 41^{5} + \left(20 a^{3} + 35 a^{2} + 28 a + 12\right)\cdot 41^{6} + \left(32 a^{3} + 2 a^{2} + 34 a + 38\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 11 }$ $=$ \( 18 a^{3} + 4 a^{2} + 24 a + 30 + \left(2 a^{3} + 10 a^{2} + 9 a + 12\right)\cdot 41 + \left(23 a^{3} + 33 a^{2} + 37 a + 10\right)\cdot 41^{2} + \left(15 a^{3} + 34 a^{2} + 17 a + 18\right)\cdot 41^{3} + \left(32 a^{3} + 5 a^{2} + 20 a + 22\right)\cdot 41^{4} + \left(38 a^{3} + 24 a^{2} + 20 a + 20\right)\cdot 41^{5} + \left(21 a^{3} + 6 a^{2} + 9 a + 34\right)\cdot 41^{6} + \left(19 a^{3} + 35 a^{2} + 26 a + 29\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display
$r_{ 12 }$ $=$ \( 3 a^{3} + 22 a^{2} + 4 a + 40 + \left(35 a^{3} + 2 a^{2} + 27 a + 9\right)\cdot 41 + \left(6 a^{3} + 22 a^{2} + 8 a + 9\right)\cdot 41^{2} + \left(23 a^{3} + 14 a^{2} + 5 a + 18\right)\cdot 41^{3} + \left(40 a^{3} + 2 a^{2} + 27 a\right)\cdot 41^{4} + \left(29 a^{3} + 12 a^{2} + 23 a + 22\right)\cdot 41^{5} + \left(15 a^{3} + 30 a^{2} + 32 a + 24\right)\cdot 41^{6} + \left(33 a^{3} + 19 a^{2} + 13 a + 37\right)\cdot 41^{7} +O(41^{8})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.