Properties

Label 1.2e2_11.10t1.1c2
Dimension 1
Group $C_{10}$
Conductor $ 2^{2} \cdot 11 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$1$
Group:$C_{10}$
Conductor:$44= 2^{2} \cdot 11 $
Artin number field: Splitting field of $f= x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_{10}$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{44}(31,\cdot)\)

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

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

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

Character values on conjugacy classes

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