Properties

Label 1.2e2_11_23.10t1.1c4
Dimension 1
Group $C_{10}$
Conductor $ 2^{2} \cdot 11 \cdot 23 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$1$
Group:$C_{10}$
Conductor:$1012= 2^{2} \cdot 11 \cdot 23 $
Artin number field: Splitting field of $f= x^{10} - 2 x^{9} - 122 x^{8} + 198 x^{7} + 5582 x^{6} - 6794 x^{5} - 118785 x^{4} + 94556 x^{3} + 1168426 x^{2} - 446344 x - 4240279 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_{10}$
Parity: Even
Corresponding Dirichlet character: \(\chi_{1012}(367,\cdot)\)

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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