Properties

Label 1.3_29.14t1.1c2
Dimension 1
Group $C_{14}$
Conductor $ 3 \cdot 29 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_{14}$
Conductor:$87= 3 \cdot 29 $
Artin number field: Splitting field of $f= x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681 $ over $\Q$
Size of Galois orbit: 6
Smallest containing permutation representation: $C_{14}$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{87}(71,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{7} + 12 x + 42 $
Roots:
$r_{ 1 }$ $=$ $ a^{6} + 18 a^{5} + 4 a^{4} + 4 a^{3} + 11 a^{2} + 34 a + 25 + \left(22 a^{6} + 45 a^{4} + 13 a^{3} + 28 a^{2} + 12 a + 21\right)\cdot 47 + \left(35 a^{6} + 23 a^{5} + 32 a^{4} + 17 a^{3} + 29 a^{2} + 31 a + 23\right)\cdot 47^{2} + \left(36 a^{6} + 23 a^{5} + 10 a^{4} + 39 a^{3} + 7 a^{2} + 4 a + 36\right)\cdot 47^{3} + \left(2 a^{6} + 29 a^{5} + 9 a^{4} + 22 a^{3} + 4 a^{2} + 21 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a^{6} + 44 a^{5} + 17 a^{4} + 41 a^{3} + 39 a^{2} + 27 a + 42 + \left(35 a^{6} + 37 a^{5} + 28 a^{4} + 18 a^{3} + 44 a^{2} + 21 a + 27\right)\cdot 47 + \left(15 a^{6} + 11 a^{5} + 29 a^{4} + 9 a^{3} + 6 a^{2} + 2 a + 8\right)\cdot 47^{2} + \left(23 a^{6} + a^{5} + 2 a^{4} + 23 a^{3} + 43 a^{2} + 15 a + 46\right)\cdot 47^{3} + \left(12 a^{6} + 32 a^{5} + 5 a^{4} + 27 a^{3} + 7 a^{2} + 42 a + 36\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 4 a^{6} + 12 a^{5} + 41 a^{4} + 28 a^{3} + 45 a^{2} + 45 a + 29 + \left(21 a^{6} + 31 a^{5} + 43 a^{4} + 23 a^{3} + 6 a^{2} + 28 a + 18\right)\cdot 47 + \left(2 a^{6} + 14 a^{5} + 12 a^{4} + 35 a^{3} + 28 a^{2} + 46 a + 46\right)\cdot 47^{2} + \left(25 a^{6} + 18 a^{5} + 14 a^{4} + 20 a^{3} + 46 a^{2} + 34 a + 36\right)\cdot 47^{3} + \left(2 a^{6} + 45 a^{5} + 7 a^{4} + 30 a^{3} + 19 a^{2} + 19 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a^{6} + 33 a^{5} + 16 a^{4} + 11 a^{3} + 21 a^{2} + 24 a + 29 + \left(41 a^{6} + 17 a^{5} + 32 a^{4} + 34 a^{3} + 5 a^{2} + 33 a + 29\right)\cdot 47 + \left(16 a^{6} + 35 a^{5} + 13 a^{4} + 18 a^{3} + 44 a^{2} + 38 a + 33\right)\cdot 47^{2} + \left(10 a^{6} + 12 a^{5} + 29 a^{4} + 32 a^{3} + 6 a^{2} + 7 a + 6\right)\cdot 47^{3} + \left(9 a^{6} + 31 a^{4} + 7 a^{3} + 9 a^{2} + 37 a + 30\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 8 a^{6} + 11 a^{5} + 27 a^{4} + 24 a^{3} + 30 a^{2} + 2 a + 14 + \left(21 a^{6} + a^{5} + 20 a^{4} + 8 a^{3} + 21 a^{2} + 38 a + 33\right)\cdot 47 + \left(46 a^{6} + 34 a^{5} + 7 a^{4} + 20 a^{3} + 30 a^{2} + 19 a + 6\right)\cdot 47^{2} + \left(8 a^{6} + 30 a^{5} + 36 a^{4} + 21 a^{3} + 46 a^{2} + 25 a + 31\right)\cdot 47^{3} + \left(9 a^{6} + 4 a^{5} + 42 a^{4} + 45 a^{3} + 36 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 22 a^{6} + 41 a^{5} + 28 a^{4} + 6 a^{3} + 44 a^{2} + 25 a + 17 + \left(44 a^{6} + 3 a^{5} + 16 a^{4} + 17 a^{3} + 13 a^{2} + 12 a + 4\right)\cdot 47 + \left(29 a^{6} + 43 a^{5} + 7 a^{4} + 19 a^{3} + 14 a^{2} + 17 a + 45\right)\cdot 47^{2} + \left(43 a^{6} + 39 a^{5} + 35 a^{4} + 41 a^{3} + 30 a^{2} + 9 a + 24\right)\cdot 47^{3} + \left(36 a^{6} + 45 a^{5} + 43 a^{4} + 26 a^{3} + 43 a^{2} + 21 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 25 a^{6} + 12 a^{5} + 21 a^{4} + 31 a^{3} + 45 a^{2} + 9 a + 10 + \left(25 a^{6} + 43 a^{5} + 37 a^{4} + 32 a^{3} + 10 a^{2} + 45 a + 44\right)\cdot 47 + \left(44 a^{6} + 9 a^{5} + 41 a^{4} + 9 a^{3} + 8 a^{2} + 45 a + 42\right)\cdot 47^{2} + \left(29 a^{6} + a^{5} + 39 a^{4} + 21 a^{3} + 23 a^{2} + 14 a + 26\right)\cdot 47^{3} + \left(28 a^{6} + 41 a^{5} + 17 a^{4} + a^{3} + 32 a^{2} + 13 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 25 a^{6} + 20 a^{5} + 33 a^{4} + 16 a^{3} + 22 a^{2} + 15 a + 10 + \left(5 a^{6} + 13 a^{5} + 13 a^{4} + 38 a^{3} + 14 a^{2} + 38 a + 33\right)\cdot 47 + \left(22 a^{6} + 42 a^{5} + 22 a^{4} + 29 a^{3} + 37 a^{2} + 17 a + 13\right)\cdot 47^{2} + \left(35 a^{6} + 31 a^{5} + 4 a^{4} + 15 a^{3} + 37 a^{2} + 32 a + 23\right)\cdot 47^{3} + \left(31 a^{6} + 33 a^{5} + 15 a^{4} + 2 a^{3} + 34 a^{2} + 5 a + 13\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 9 }$ $=$ $ 25 a^{6} + 38 a^{5} + 15 a^{4} + 29 a^{3} + 6 a^{2} + 23 a + 21 + \left(37 a^{6} + 11 a^{5} + 7 a^{4} + 40 a^{3} + 14 a^{2} + 2 a + 40\right)\cdot 47 + \left(17 a^{6} + 8 a^{5} + 38 a^{4} + 2 a^{3} + 45 a^{2} + 20\right)\cdot 47^{2} + \left(40 a^{6} + 33 a^{5} + 41 a^{4} + a^{3} + 38 a^{2} + 33 a + 38\right)\cdot 47^{3} + \left(17 a^{6} + 18 a^{5} + 21 a^{4} + 34 a^{3} + 25 a^{2} + 15 a + 26\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 10 }$ $=$ $ 26 a^{6} + 3 a^{5} + 39 a^{4} + 30 a^{3} + 46 a^{2} + 39 a + 38 + \left(29 a^{6} + 38 a^{5} + 15 a^{4} + a^{3} + 3 a^{2} + 10 a + 18\right)\cdot 47 + \left(5 a^{6} + 11 a^{5} + 33 a^{4} + 17 a^{3} + 4 a^{2} + 34 a + 43\right)\cdot 47^{2} + \left(26 a^{5} + 31 a^{4} + a^{3} + 39 a^{2} + 44 a + 33\right)\cdot 47^{3} + \left(45 a^{5} + 6 a^{3} + 9 a^{2} + 8 a + 17\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 11 }$ $=$ $ 30 a^{6} + a^{5} + 42 a^{4} + 24 a^{3} + a^{2} + 14 a + 12 + \left(35 a^{6} + 16 a^{5} + 5 a^{4} + 27 a^{3} + 30 a^{2} + 5 a + 14\right)\cdot 47 + \left(12 a^{6} + 41 a^{5} + 45 a^{4} + 24 a^{3} + 41 a^{2} + 31 a + 9\right)\cdot 47^{2} + \left(36 a^{6} + 31 a^{5} + 42 a^{4} + 34 a^{3} + 12 a^{2} + 45 a + 43\right)\cdot 47^{3} + \left(32 a^{6} + 18 a^{5} + 40 a^{4} + 7 a^{3} + 41 a^{2} + 21 a + 45\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 12 }$ $=$ $ 33 a^{6} + 2 a^{5} + 9 a^{4} + 10 a^{3} + 5 a^{2} + 34 a + 5 + \left(37 a^{6} + 44 a^{5} + 34 a^{4} + 27 a^{3} + 30 a^{2} + 7 a + 35\right)\cdot 47 + \left(3 a^{6} + 3 a^{5} + 34 a^{4} + 20 a^{3} + 33 a^{2} + 5 a + 26\right)\cdot 47^{2} + \left(27 a^{6} + 5 a^{5} + 39 a^{4} + 35 a^{3} + 22 a^{2} + 31 a + 17\right)\cdot 47^{3} + \left(6 a^{6} + 6 a^{5} + 7 a^{4} + a^{3} + 32 a^{2} + a + 16\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 13 }$ $=$ $ 37 a^{6} + 28 a^{5} + 10 a^{4} + 34 a^{3} + a^{2} + 38 a + 37 + \left(10 a^{6} + 35 a^{5} + 2 a^{4} + 20 a^{3} + 22 a^{2} + 26 a + 13\right)\cdot 47 + \left(26 a^{6} + 8 a^{5} + 2 a^{4} + 13 a^{3} + 24 a^{2} + 33 a + 20\right)\cdot 47^{2} + \left(20 a^{6} + 21 a^{5} + 3 a^{4} + 4 a^{3} + 23 a^{2} + 28 a + 2\right)\cdot 47^{3} + \left(34 a^{6} + 17 a^{5} + 34 a^{4} + 33 a^{3} + 40 a^{2} + 23 a + 16\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 14 }$ $=$ $ 40 a^{6} + 19 a^{5} + 27 a^{4} + 41 a^{3} + 13 a^{2} + 41 + \left(8 a^{6} + 34 a^{5} + 25 a^{4} + 24 a^{3} + 35 a^{2} + 45 a + 40\right)\cdot 47 + \left(2 a^{6} + 40 a^{5} + 7 a^{4} + 43 a^{3} + 27 a^{2} + 4 a + 34\right)\cdot 47^{2} + \left(38 a^{6} + 4 a^{5} + 44 a^{4} + 36 a^{3} + 43 a^{2} + a + 7\right)\cdot 47^{3} + \left(9 a^{6} + 37 a^{5} + 3 a^{4} + 34 a^{3} + 25 a^{2} + 13 a + 11\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 14 }$

Cycle notation
$(1,5)(2,13)(3,11)(4,14)(6,7)(8,10)(9,12)$
$(1,2,12,3,8,7,4)(5,13,9,11,10,6,14)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 14 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,5)(2,13)(3,11)(4,14)(6,7)(8,10)(9,12)$$-1$
$1$$7$$(1,2,12,3,8,7,4)(5,13,9,11,10,6,14)$$\zeta_{7}^{2}$
$1$$7$$(1,12,8,4,2,3,7)(5,9,10,14,13,11,6)$$\zeta_{7}^{4}$
$1$$7$$(1,3,4,12,7,2,8)(5,11,14,9,6,13,10)$$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$
$1$$7$$(1,8,2,7,12,4,3)(5,10,13,6,9,14,11)$$\zeta_{7}$
$1$$7$$(1,7,3,2,4,8,12)(5,6,11,13,14,10,9)$$\zeta_{7}^{3}$
$1$$7$$(1,4,7,8,3,12,2)(5,14,6,10,11,9,13)$$\zeta_{7}^{5}$
$1$$14$$(1,13,12,11,8,6,4,5,2,9,3,10,7,14)$$-\zeta_{7}^{2}$
$1$$14$$(1,11,4,9,7,13,8,5,3,14,12,6,2,10)$$\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$
$1$$14$$(1,6,3,13,4,10,12,5,7,11,2,14,8,9)$$-\zeta_{7}^{3}$
$1$$14$$(1,9,8,14,2,11,7,5,12,10,4,13,3,6)$$-\zeta_{7}^{4}$
$1$$14$$(1,10,2,6,12,14,3,5,8,13,7,9,4,11)$$-\zeta_{7}$
$1$$14$$(1,14,7,10,3,9,2,5,4,6,8,11,12,13)$$-\zeta_{7}^{5}$
The blue line marks the conjugacy class containing complex conjugation.