Properties

Label 2.439.15t2.a.c
Dimension $2$
Group $D_{15}$
Conductor $439$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{15}$
Conductor: \(439\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 15.1.3142328914862177479.1
Galois orbit size: $4$
Smallest permutation container: $D_{15}$
Parity: odd
Determinant: 1.439.2t1.a.a
Projective image: $D_{15}$
Projective stem field: 15.1.3142328914862177479.1

Defining polynomial

$f(x)$$=$\(x^{15} - 5 x^{14} + 11 x^{13} - 9 x^{12} - 7 x^{11} + 17 x^{10} - x^{9} - 29 x^{8} + 38 x^{7} - 13 x^{6} - 20 x^{5} + 24 x^{4} + 7 x^{3} - 23 x^{2} + 13 x - 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \(x^{5} + 5 x + 17\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a^{3} + 18 a^{2} + 12 a + 9 + \left(11 a^{4} + 4 a^{3} + 3 a^{2} + 6 a + 7\right)\cdot 19 + \left(10 a^{4} + 16 a^{3} + 7 a^{2} + 13 a + 5\right)\cdot 19^{2} + \left(10 a^{4} + 15 a^{3} + 5 a^{2} + 13 a + 14\right)\cdot 19^{3} + \left(17 a^{4} + 3 a^{3} + 6 a^{2} + 4 a + 14\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( a^{4} + 4 a^{3} + 12 a^{2} + 7 a + 10 + \left(2 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 19 + \left(14 a^{4} + 14 a^{3} + a^{2} + 15 a + 17\right)\cdot 19^{2} + \left(2 a^{4} + 8 a^{3} + 17 a^{2} + 14 a + 17\right)\cdot 19^{3} + \left(14 a^{4} + 10 a^{3} + 12 a^{2} + 4 a + 2\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 3 a^{4} + 15 a^{3} + 14 a^{2} + 9 a + 2 + \left(13 a^{4} + 7 a^{3} + 12 a + 16\right)\cdot 19 + \left(18 a^{4} + 5 a^{2} + 11 a + 18\right)\cdot 19^{2} + \left(13 a^{4} + 2 a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 19^{3} + \left(14 a^{4} + 13 a^{3} + 15 a^{2} + 5 a + 3\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 4 a^{4} + 10 a^{3} + 15 a^{2} + 16 a + 3 + \left(8 a^{4} + 6 a^{3} + 10 a^{2} + 1\right)\cdot 19 + \left(4 a^{4} + 8 a^{3} + 10 a^{2} + 9 a + 17\right)\cdot 19^{2} + \left(3 a^{4} + 9 a^{3} + 18 a^{2} + 7 a\right)\cdot 19^{3} + \left(12 a^{4} + 7 a^{3} + 9 a^{2} + 6 a + 14\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 5 a^{4} + 18 a^{3} + 12 a^{2} + 5 a + 6 + \left(a^{4} + a^{3} + 18 a + 16\right)\cdot 19 + \left(12 a^{4} + 4 a^{3} + 7 a^{2} + 13 a + 9\right)\cdot 19^{2} + \left(4 a^{4} + 15 a^{3} + 11 a^{2} + 11 a + 1\right)\cdot 19^{3} + \left(7 a^{4} + 2 a^{3} + 4 a^{2} + 10 a + 5\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 7 a^{4} + 16 a^{2} + 9 a + 15 + \left(9 a^{4} + 15 a^{3} + 14 a^{2} + 15 a + 5\right)\cdot 19 + \left(11 a^{4} + 14 a^{3} + 7 a^{2} + 5 a + 7\right)\cdot 19^{2} + \left(10 a^{4} + 12 a^{3} + 7 a^{2} + 9 a + 11\right)\cdot 19^{3} + \left(14 a^{4} + 3 a^{3} + 17 a^{2} + 4\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 8 a^{4} + a^{3} + 8 a^{2} + 3 + \left(15 a^{4} + 18 a^{3} + 11 a^{2} + 16 a + 6\right)\cdot 19 + \left(17 a^{4} + 16 a^{3} + 5 a^{2} + 12 a + 15\right)\cdot 19^{2} + \left(6 a^{4} + 11 a^{3} + 15 a^{2} + 17 a + 18\right)\cdot 19^{3} + \left(18 a^{4} + 8 a^{3} + 7 a^{2} + 15 a + 17\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 8 a^{4} + 8 a^{3} + 4 a^{2} + 9 a + \left(13 a^{4} + 4 a^{3} + a^{2} + 16 a + 3\right)\cdot 19 + \left(10 a^{4} + 11 a^{3} + 10 a^{2} + 7 a + 4\right)\cdot 19^{2} + \left(9 a^{4} + 8 a^{3} + 13 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(5 a^{4} + 2 a^{3} + 6 a^{2} + 2 a + 6\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 10 a^{4} + a^{3} + 18 a^{2} + 9 a + 7 + \left(15 a^{4} + 12 a^{3} + 11 a^{2} + 5 a + 16\right)\cdot 19 + \left(18 a^{4} + 17 a + 17\right)\cdot 19^{2} + \left(5 a^{4} + 15 a^{3} + 3 a^{2} + 14 a + 6\right)\cdot 19^{3} + \left(17 a^{4} + 3 a^{3} + 12 a + 7\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 12 a^{4} + 17 a^{3} + 16 a^{2} + \left(12 a^{4} + 14 a^{3} + 2 a^{2} + 8 a + 14\right)\cdot 19 + \left(9 a^{4} + 3 a^{3} + 4 a^{2} + 1\right)\cdot 19^{2} + \left(12 a^{4} + 8 a^{3} + 8 a^{2} + 13 a + 3\right)\cdot 19^{3} + \left(16 a^{4} + 4 a^{3} + 6 a^{2} + 16 a + 11\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 11 }$ $=$ \( 13 a^{4} + 7 a^{3} + 18 a + \left(17 a^{4} + 3 a^{3} + 14 a^{2} + 11 a + 6\right)\cdot 19 + \left(15 a^{4} + 7 a^{3} + 11 a^{2} + 17 a + 6\right)\cdot 19^{2} + \left(6 a^{4} + 17 a^{2} + 5 a + 10\right)\cdot 19^{3} + \left(13 a^{4} + 11 a^{3} + 13 a^{2} + 4 a + 10\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 12 }$ $=$ \( 13 a^{4} + 13 a^{3} + 13 a^{2} + 16 a + \left(9 a^{4} + 11 a^{3} + 13 a^{2} + 16 a + 12\right)\cdot 19 + \left(6 a^{4} + 10 a^{3} + 9 a^{2} + 3 a + 6\right)\cdot 19^{2} + \left(3 a^{4} + 10 a^{3} + a^{2} + 3 a + 15\right)\cdot 19^{3} + \left(16 a^{4} + a^{3} + 5 a^{2} + 17 a + 2\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 13 }$ $=$ \( 15 a^{4} + 16 a^{3} + a^{2} + 17 a + 12 + \left(4 a^{4} + 11 a^{3} + 13 a + 1\right)\cdot 19 + \left(16 a^{2} + 18 a + 2\right)\cdot 19^{2} + \left(13 a^{4} + 2 a^{2} + 6 a + 5\right)\cdot 19^{3} + \left(8 a^{4} + 8 a^{3} + 2 a^{2} + 14 a + 17\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 14 }$ $=$ \( 16 a^{4} + 18 a^{3} + 14 a^{2} + 9 a + 12 + \left(12 a^{4} + 8 a^{3} + 16 a^{2} + 4 a + 5\right)\cdot 19 + \left(3 a^{4} + 15 a^{3} + 8 a^{2} + 4 a + 14\right)\cdot 19^{2} + \left(17 a^{4} + 15 a^{3} + 4 a^{2} + 2 a + 13\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 14 a^{2} + 12 a + 6\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 15 }$ $=$ \( 18 a^{4} + 16 a^{3} + 10 a^{2} + 16 a + 2 + \left(6 a^{4} + 9 a^{3} + 4 a^{2} + 11 a + 15\right)\cdot 19 + \left(16 a^{4} + 8 a^{3} + 8 a^{2} + 18 a + 7\right)\cdot 19^{2} + \left(11 a^{4} + 17 a^{3} + 18 a + 16\right)\cdot 19^{3} + \left(10 a^{4} + 13 a^{3} + 10 a^{2} + 4 a + 7\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

Cycle notation
$(1,14,2)(3,11,6)(4,7,12)(5,15,10)(8,13,9)$
$(1,13,3,7,10)(2,8,6,4,15)(5,14,9,11,12)$
$(2,14)(3,7)(4,11)(5,8)(6,12)(9,15)(10,13)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 15 }$ Character value
$1$$1$$()$$2$
$15$$2$$(2,14)(3,7)(4,11)(5,8)(6,12)(9,15)(10,13)$$0$
$2$$3$$(1,14,2)(3,11,6)(4,7,12)(5,15,10)(8,13,9)$$-1$
$2$$5$$(1,13,3,7,10)(2,8,6,4,15)(5,14,9,11,12)$$\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$
$2$$5$$(1,3,10,13,7)(2,6,15,8,4)(5,9,12,14,11)$$-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$
$2$$15$$(1,9,6,7,5,2,13,11,4,10,14,8,3,12,15)$$-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$
$2$$15$$(1,6,5,13,4,14,3,15,9,7,2,11,10,8,12)$$2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$
$2$$15$$(1,5,4,3,9,2,10,12,6,13,14,15,7,11,8)$$-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$
$2$$15$$(1,11,15,13,12,2,3,5,8,7,14,6,10,9,4)$$-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$

The blue line marks the conjugacy class containing complex conjugation.