Properties

Label 2.3639.10t3.b.a
Dimension $2$
Group $D_{10}$
Conductor $3639$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{10}$
Conductor: \(3639\)\(\medspace = 3 \cdot 1213 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 10.2.212710546411520733.1
Galois orbit size: $2$
Smallest permutation container: $D_{10}$
Parity: odd
Determinant: 1.3639.2t1.a.a
Projective image: $D_5$
Projective stem field: 5.1.13242321.1

Defining polynomial

$f(x)$$=$\(x^{10} - 2 x^{9} + 17 x^{8} - 80 x^{7} + 131 x^{6} - 230 x^{5} + 763 x^{4} + 2184 x^{3} - 11847 x^{2} + 16803 x - 4257\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{5} + 4 x + 11\)  Toggle raw display

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,6)(2,7)(3,8)(4,9)(5,10)$$-2$
$5$$2$$(1,7)(2,6)(3,10)(5,8)$$0$
$5$$2$$(1,2)(3,5)(4,9)(6,7)(8,10)$$0$
$2$$5$$(1,7,8,4,5)(2,3,9,10,6)$$\zeta_{5}^{3} + \zeta_{5}^{2}$
$2$$5$$(1,8,5,7,4)(2,9,6,3,10)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$$10$$(1,2,8,9,5,6,7,3,4,10)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
$2$$10$$(1,9,7,10,8,6,4,2,5,3)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$

The blue line marks the conjugacy class containing complex conjugation.