Properties

Label 1.100.10t1.a.a
Dimension 1
Group $C_{10}$
Conductor $ 2^{2} \cdot 5^{2}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_{10}$
Conductor:$100= 2^{2} \cdot 5^{2} $
Artin number field: Splitting field of 10.0.156250000000000.1 defined by $f= x^{10} + 20 x^{8} + 120 x^{6} + 225 x^{4} + 90 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_{10}$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{100}(11,\cdot)\)
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 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{5} + 4 x + 11 $
Roots:
$r_{ 1 }$ $=$ $ 4 a^{4} + a^{3} + 2 a^{2} + 6 a + 5 + \left(10 a^{4} + 4 a^{3} + 8 a^{2} + 3 a + 4\right)\cdot 13 + \left(6 a^{3} + 7 a^{2} + 2 a + 10\right)\cdot 13^{2} + \left(a^{4} + 2 a^{3} + 12 a^{2} + 7 a\right)\cdot 13^{3} + \left(a^{4} + 8 a^{3} + 10 a^{2} + 4 a + 6\right)\cdot 13^{4} + \left(4 a^{4} + 8 a^{3} + 6 a^{2} + 3 a + 10\right)\cdot 13^{5} + \left(5 a^{4} + 5 a^{3} + 10 a^{2} + 5 a + 11\right)\cdot 13^{6} + \left(8 a^{4} + 3 a^{3} + a\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 9 a^{4} + 10 a^{3} + 8 a^{2} + 8 + \left(2 a^{4} + 11 a^{3} + a^{2} + 5 a + 8\right)\cdot 13 + \left(7 a^{4} + 7 a^{3} + 6 a^{2} + 4 a + 12\right)\cdot 13^{2} + \left(12 a^{4} + 6 a^{3} + 7 a^{2} + 2 a + 8\right)\cdot 13^{3} + \left(7 a^{4} + 4 a^{3} + 2 a + 4\right)\cdot 13^{4} + \left(12 a^{4} + 3 a^{3} + 6 a^{2} + 6 a + 1\right)\cdot 13^{5} + \left(4 a^{4} + 2 a^{3} + 12 a^{2} + 4 a + 8\right)\cdot 13^{6} + \left(3 a^{4} + 4 a^{2} + 11 a + 5\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 3 a^{4} + 10 a^{3} + 4 a + 7 + \left(9 a^{4} + 10 a^{3} + 7 a^{2} + 7 a + 3\right)\cdot 13 + \left(11 a^{4} + 6 a^{3} + 2 a^{2} + 2 a + 6\right)\cdot 13^{2} + \left(5 a^{4} + 6 a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 13^{3} + \left(6 a^{4} + 8 a^{3} + 5 a^{2} + 7\right)\cdot 13^{4} + \left(6 a^{4} + 3 a^{3} + 11 a^{2} + 4 a + 2\right)\cdot 13^{5} + \left(a^{4} + 7 a^{3} + 3 a^{2} + 9 a + 2\right)\cdot 13^{6} + \left(6 a^{4} + 12 a^{3} + a^{2} + 11 a + 9\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 11 a^{4} + 3 a^{2} + 4 + \left(10 a^{4} + 12 a^{3} + 2 a^{2} + 6 a + 11\right)\cdot 13 + \left(3 a^{4} + 7 a^{3} + 11 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(12 a^{3} + a^{2} + 5 a + 6\right)\cdot 13^{3} + \left(4 a^{4} + 3 a^{3} + 3 a^{2} + 2\right)\cdot 13^{4} + \left(11 a^{4} + 5 a^{3} + 7 a^{2} + 4 a + 10\right)\cdot 13^{5} + \left(a^{4} + 10 a^{3} + 9 a^{2} + 11 a\right)\cdot 13^{6} + \left(12 a^{4} + 6 a^{3} + 7 a^{2} + 4 a + 5\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 5 a^{4} + 8 a^{3} + 7 a^{2} + 10 a + 3 + \left(11 a^{4} + a^{3} + a^{2} + 9 a + 5\right)\cdot 13 + \left(2 a^{4} + 5 a^{2} + 3 a + 1\right)\cdot 13^{2} + \left(6 a^{4} + 3 a^{3} + 11 a^{2} + 9 a + 12\right)\cdot 13^{3} + \left(11 a^{4} + 4 a^{3} + 6 a + 2\right)\cdot 13^{4} + \left(11 a^{4} + 10 a^{3} + 4 a^{2} + 9 a + 4\right)\cdot 13^{5} + \left(9 a^{4} + 4 a^{3} + 4 a^{2} + 7 a + 8\right)\cdot 13^{6} + \left(5 a^{4} + 9 a^{3} + 12 a^{2} + 6 a + 10\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 9 a^{4} + 12 a^{3} + 11 a^{2} + 7 a + 8 + \left(2 a^{4} + 8 a^{3} + 4 a^{2} + 9 a + 8\right)\cdot 13 + \left(12 a^{4} + 6 a^{3} + 5 a^{2} + 10 a + 2\right)\cdot 13^{2} + \left(11 a^{4} + 10 a^{3} + 5 a + 12\right)\cdot 13^{3} + \left(11 a^{4} + 4 a^{3} + 2 a^{2} + 8 a + 6\right)\cdot 13^{4} + \left(8 a^{4} + 4 a^{3} + 6 a^{2} + 9 a + 2\right)\cdot 13^{5} + \left(7 a^{4} + 7 a^{3} + 2 a^{2} + 7 a + 1\right)\cdot 13^{6} + \left(4 a^{4} + 9 a^{3} + 12 a^{2} + 11 a + 12\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 4 a^{4} + 3 a^{3} + 5 a^{2} + 5 + \left(10 a^{4} + a^{3} + 11 a^{2} + 8 a + 4\right)\cdot 13 + \left(5 a^{4} + 5 a^{3} + 6 a^{2} + 8 a\right)\cdot 13^{2} + \left(6 a^{3} + 5 a^{2} + 10 a + 4\right)\cdot 13^{3} + \left(5 a^{4} + 8 a^{3} + 12 a^{2} + 10 a + 8\right)\cdot 13^{4} + \left(9 a^{3} + 6 a^{2} + 6 a + 11\right)\cdot 13^{5} + \left(8 a^{4} + 10 a^{3} + 8 a + 4\right)\cdot 13^{6} + \left(9 a^{4} + 12 a^{3} + 8 a^{2} + a + 7\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 8 }$ $=$ $ 10 a^{4} + 3 a^{3} + 9 a + 6 + \left(3 a^{4} + 2 a^{3} + 6 a^{2} + 5 a + 9\right)\cdot 13 + \left(a^{4} + 6 a^{3} + 10 a^{2} + 10 a + 6\right)\cdot 13^{2} + \left(7 a^{4} + 6 a^{3} + 6 a^{2} + 7 a + 4\right)\cdot 13^{3} + \left(6 a^{4} + 4 a^{3} + 7 a^{2} + 12 a + 5\right)\cdot 13^{4} + \left(6 a^{4} + 9 a^{3} + a^{2} + 8 a + 10\right)\cdot 13^{5} + \left(11 a^{4} + 5 a^{3} + 9 a^{2} + 3 a + 10\right)\cdot 13^{6} + \left(6 a^{4} + 11 a^{2} + a + 3\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 9 }$ $=$ $ 2 a^{4} + 10 a^{2} + 9 + \left(2 a^{4} + a^{3} + 10 a^{2} + 7 a + 1\right)\cdot 13 + \left(9 a^{4} + 5 a^{3} + a^{2} + 7 a + 11\right)\cdot 13^{2} + \left(12 a^{4} + 11 a^{2} + 7 a + 6\right)\cdot 13^{3} + \left(8 a^{4} + 9 a^{3} + 9 a^{2} + 12 a + 10\right)\cdot 13^{4} + \left(a^{4} + 7 a^{3} + 5 a^{2} + 8 a + 2\right)\cdot 13^{5} + \left(11 a^{4} + 2 a^{3} + 3 a^{2} + a + 12\right)\cdot 13^{6} + \left(6 a^{3} + 5 a^{2} + 8 a + 7\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 10 }$ $=$ $ 8 a^{4} + 5 a^{3} + 6 a^{2} + 3 a + 10 + \left(a^{4} + 11 a^{3} + 11 a^{2} + 3 a + 7\right)\cdot 13 + \left(10 a^{4} + 12 a^{3} + 7 a^{2} + 9 a + 11\right)\cdot 13^{2} + \left(6 a^{4} + 9 a^{3} + a^{2} + 3 a\right)\cdot 13^{3} + \left(a^{4} + 8 a^{3} + 12 a^{2} + 6 a + 10\right)\cdot 13^{4} + \left(a^{4} + 2 a^{3} + 8 a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(3 a^{4} + 8 a^{3} + 8 a^{2} + 5 a + 4\right)\cdot 13^{6} + \left(7 a^{4} + 3 a^{3} + 6 a + 2\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$

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

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

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