Properties

Label 2.100.10t6.b.d
Dimension 2
Group $D_5\times C_5$
Conductor $ 2^{2} \cdot 5^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$D_5\times C_5$
Conductor:$100= 2^{2} \cdot 5^{2} $
Artin number field: Splitting field of 10.0.400000000.1 defined by $f= x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $D_5\times C_5$
Parity: Odd
Determinant: 1.100.10t1.a.d
Projective image: $D_5$
Projective field: Galois closure of 5.1.6250000.1

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character value
$1$$1$$()$$2$
$5$$2$$(1,2)(3,6)(4,10)(5,9)(7,8)$$0$
$1$$5$$(1,7,6,4,5)(2,8,3,10,9)$$2 \zeta_{5}$
$1$$5$$(1,6,5,7,4)(2,3,9,8,10)$$2 \zeta_{5}^{2}$
$1$$5$$(1,4,7,5,6)(2,10,8,9,3)$$2 \zeta_{5}^{3}$
$1$$5$$(1,5,4,6,7)(2,9,10,3,8)$$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$
$2$$5$$(1,7,6,4,5)(2,9,10,3,8)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$$5$$(1,6,5,7,4)(2,10,8,9,3)$$\zeta_{5}^{3} + \zeta_{5}^{2}$
$2$$5$$(2,8,3,10,9)$$\zeta_{5} + 1$
$2$$5$$(2,3,9,8,10)$$\zeta_{5}^{2} + 1$
$2$$5$$(2,10,8,9,3)$$\zeta_{5}^{3} + 1$
$2$$5$$(2,9,10,3,8)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$
$2$$5$$(1,5,4,6,7)(2,3,9,8,10)$$-\zeta_{5}^{3} - \zeta_{5} - 1$
$2$$5$$(1,4,7,5,6)(2,9,10,3,8)$$-\zeta_{5}^{2} - \zeta_{5} - 1$
$2$$5$$(1,6,5,7,4)(2,8,3,10,9)$$\zeta_{5}^{2} + \zeta_{5}$
$2$$5$$(1,7,6,4,5)(2,10,8,9,3)$$\zeta_{5}^{3} + \zeta_{5}$
$5$$10$$(1,8,7,3,6,10,4,9,5,2)$$0$
$5$$10$$(1,3,4,2,7,10,5,8,6,9)$$0$
$5$$10$$(1,9,6,8,5,10,7,2,4,3)$$0$
$5$$10$$(1,2,5,9,4,10,6,3,7,8)$$0$
The blue line marks the conjugacy class containing complex conjugation.