Properties

Label 1.93.10t1.a.c
Dimension 1
Group $C_{10}$
Conductor $ 3 \cdot 31 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_{10}$
Conductor:$93= 3 \cdot 31 $
Artin number field: Splitting field of 10.0.207252522098163.1 defined by $f= x^{10} - x^{9} + 13 x^{8} - 30 x^{7} + 164 x^{6} - 255 x^{5} + 448 x^{4} - 99 x^{3} + 106 x^{2} - 5 x + 25 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_{10}$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{93}(35,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{5} + 4 x + 11 $
Roots:
$r_{ 1 }$ $=$ $ 9 a^{4} + 9 a^{3} + 4 a^{2} + 10 a + 10 + \left(10 a^{4} + 9 a^{3} + 5 a^{2} + 8 a\right)\cdot 13 + \left(10 a^{4} + 5 a^{3} + 8 a^{2} + 10 a + 2\right)\cdot 13^{2} + \left(4 a^{4} + 12 a^{3} + 9 a^{2} + 11 a + 3\right)\cdot 13^{3} + \left(2 a^{4} + 4 a^{3} + 7 a^{2} + 10 a + 11\right)\cdot 13^{4} + \left(11 a^{4} + 8 a^{3} + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 11 a^{4} + 4 a^{3} + 8 a^{2} + 12 a + 10 + \left(7 a^{4} + 11 a^{3} + 12 a^{2} + 9 a + 11\right)\cdot 13 + \left(6 a^{4} + 11 a^{3} + 12 a^{2} + 6 a + 6\right)\cdot 13^{2} + \left(4 a^{4} + 5 a + 8\right)\cdot 13^{3} + \left(4 a^{4} + 7 a^{3} + a^{2} + 7 a + 7\right)\cdot 13^{4} + \left(6 a^{4} + 8 a^{2} + 2 a + 2\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 11 a^{3} + 5 a^{2} + a + 2 + \left(a^{4} + 3 a^{3} + 12 a^{2} + 7 a + 6\right)\cdot 13 + \left(11 a^{4} + 3 a^{2} + 2 a + 5\right)\cdot 13^{2} + \left(10 a^{4} + 11 a^{3} + 5 a^{2} + 5 a + 9\right)\cdot 13^{3} + \left(11 a^{4} + 3 a^{3} + 4 a^{2} + 12 a + 7\right)\cdot 13^{4} + \left(7 a^{4} + 10 a^{3} + 5 a^{2} + 12 a + 6\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 5 a^{4} + 7 a^{3} + 2 a^{2} + 12 a + 5 + \left(10 a^{4} + 10 a^{3} + 9 a^{2} + 5 a + 2\right)\cdot 13 + \left(a^{4} + 6 a^{3} + a^{2} + 7 a + 12\right)\cdot 13^{2} + \left(12 a^{4} + a^{3} + 10 a^{2} + 10\right)\cdot 13^{3} + \left(3 a^{4} + 7 a^{3} + 4 a^{2} + 12 a + 5\right)\cdot 13^{4} + \left(8 a^{4} + 5 a^{2} + 10 a + 10\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 5 }$ $=$ $ a^{4} + a^{3} + 12 a^{2} + 4 a + 4 + \left(a^{4} + 11 a^{3} + 7 a^{2} + 6 a + 8\right)\cdot 13 + \left(9 a^{4} + 8 a^{3} + 3 a^{2} + 6 a + 9\right)\cdot 13^{2} + \left(5 a^{4} + 5 a^{3} + 7 a^{2} + 9 a + 9\right)\cdot 13^{3} + \left(11 a^{4} + 11 a^{3} + 8 a^{2} + 11 a + 1\right)\cdot 13^{4} + \left(5 a^{4} + 5 a^{3} + 5 a^{2} + 6 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 4 a^{4} + 2 a^{3} + 8 a^{2} + 12 a + 7 + \left(7 a^{2} + 6 a + 6\right)\cdot 13 + \left(12 a^{4} + a^{3} + 3 a^{2} + 5 a + 8\right)\cdot 13^{2} + \left(5 a^{3} + 9 a^{2} + 12 a + 3\right)\cdot 13^{3} + \left(2 a^{4} + 2 a^{3} + 11 a^{2} + 8 a + 10\right)\cdot 13^{4} + \left(a^{4} + 12 a^{2} + 7 a + 5\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 7 a^{3} + 2 a^{2} + 3 a + 6 + \left(3 a^{4} + 2 a^{3} + a^{2} + 6 a + 1\right)\cdot 13 + \left(5 a^{4} + 5 a^{3} + 9 a^{2} + 5\right)\cdot 13^{2} + \left(4 a^{4} + 2 a^{3} + 9 a^{2} + 4 a\right)\cdot 13^{3} + \left(9 a^{4} + 12 a^{3} + 7 a^{2} + 11 a + 8\right)\cdot 13^{4} + \left(8 a^{3} + 7 a^{2} + 7 a + 10\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 2 a^{4} + 8 a^{3} + 6 a^{2} + 10 a + 2 + \left(8 a^{4} + 4 a^{3} + 10 a^{2} + 6 a + 10\right)\cdot 13 + \left(6 a^{4} + 9 a^{3} + 3 a + 1\right)\cdot 13^{2} + \left(7 a^{4} + 10 a^{3} + 5 a^{2} + 6 a + 5\right)\cdot 13^{3} + \left(11 a^{4} + a^{3} + 11 a^{2} + 3 a + 7\right)\cdot 13^{4} + \left(5 a^{4} + 4 a^{3} + 4 a^{2} + 2 a + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 9 }$ $=$ $ 8 a^{4} + 10 a^{3} + 7 a^{2} + 4 a + 12 + \left(3 a^{4} + a^{3} + 4 a^{2} + 10 a + 11\right)\cdot 13 + \left(3 a^{4} + 12 a^{3} + 8 a^{2} + 12 a + 3\right)\cdot 13^{2} + \left(10 a^{4} + 8 a^{3} + 4 a^{2} + 8 a + 2\right)\cdot 13^{3} + \left(5 a^{4} + 7 a^{3} + 10 a^{2} + 7 a + 9\right)\cdot 13^{4} + \left(10 a^{4} + 6 a^{3} + a^{2} + 6 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 10 }$ $=$ $ 12 a^{4} + 6 a^{3} + 11 a^{2} + 10 a + 8 + \left(5 a^{4} + 9 a^{3} + 6 a^{2} + 9 a + 5\right)\cdot 13 + \left(11 a^{4} + 3 a^{3} + 12 a^{2} + 8 a + 9\right)\cdot 13^{2} + \left(3 a^{4} + 6 a^{3} + 2 a^{2} + 11\right)\cdot 13^{3} + \left(2 a^{4} + 6 a^{3} + 10 a^{2} + 5 a + 8\right)\cdot 13^{4} + \left(7 a^{4} + 6 a^{3} + 12 a^{2} + 6 a + 2\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

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

Character values on conjugacy classes

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