Properties

Label 2.3639.15t2.a.b
Dimension $2$
Group $D_{15}$
Conductor $3639$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{15}$
Conductor: \(3639\)\(\medspace = 3 \cdot 1213 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 15.1.8450344007000266933623879.1
Galois orbit size: $4$
Smallest permutation container: $D_{15}$
Parity: odd
Determinant: 1.3639.2t1.a.a
Projective image: $D_{15}$
Projective stem field: 15.1.8450344007000266933623879.1

Defining polynomial

$f(x)$$=$\(x^{15} - 6 x^{14} + 29 x^{13} - 119 x^{12} + 396 x^{11} - 1132 x^{10} + 2906 x^{9} - 6522 x^{8} + 13105 x^{7} - 24007 x^{6} + 38721 x^{5} - 52199 x^{4} + 52599 x^{3} - 37137 x^{2} + 14859 x - 2421\)  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:
$r_{ 1 }$ $=$ \( a^{4} + 4 a^{3} + 3 a^{2} + 2 a + 11 + \left(13 a^{4} + 16 a^{3} + 7 a^{2} + 2 a + 7\right)\cdot 19 + \left(15 a^{4} + 9 a^{3} + 17 a^{2} + 11 a + 17\right)\cdot 19^{2} + \left(4 a^{4} + 7 a^{3} + 10 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(18 a^{4} + 3 a^{3} + 8 a^{2} + 11 a + 2\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 a^{4} + 11 a^{3} + 16 a^{2} + 7 a + 4 + \left(8 a^{4} + 16 a^{3} + 5 a^{2} + 6 a\right)\cdot 19 + \left(2 a^{4} + 13 a^{3} + 8 a^{2} + 8 a + 15\right)\cdot 19^{2} + \left(3 a^{4} + 14 a^{3} + 5 a^{2} + 14 a + 13\right)\cdot 19^{3} + \left(13 a^{4} + 3 a^{3} + 16 a^{2} + 3 a + 16\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 2 a^{4} + 16 a^{3} + 2 a^{2} + 7 a + 4 + \left(17 a^{4} + 18 a^{3} + 4 a^{2} + 18 a + 17\right)\cdot 19 + \left(14 a^{4} + 10 a^{3} + 18 a^{2} + 15 a + 7\right)\cdot 19^{2} + \left(17 a^{4} + 11 a^{3} + 18 a^{2} + 16 a + 15\right)\cdot 19^{3} + \left(17 a^{4} + 10 a^{3} + 5 a^{2} + 14 a + 16\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 4 a^{4} + 4 a^{3} + 13 a^{2} + 5 a + 12 + \left(a^{4} + 7 a^{3} + 16 a^{2} + 15 a + 10\right)\cdot 19 + \left(12 a^{4} + 5 a^{3} + 13 a^{2} + 11 a + 15\right)\cdot 19^{2} + \left(14 a^{4} + 13 a^{3} + a^{2} + 14 a + 2\right)\cdot 19^{3} + \left(15 a^{4} + 9 a^{3} + 4 a^{2} + 18 a + 8\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 4 a^{4} + 13 a^{3} + 8 a + 18 + \left(15 a^{4} + 8 a^{3} + 3 a^{2} + 17 a\right)\cdot 19 + \left(4 a^{4} + 4 a + 6\right)\cdot 19^{2} + \left(11 a^{4} + 5 a^{3} + 16 a^{2} + 5 a + 13\right)\cdot 19^{3} + \left(5 a^{4} + 14 a^{3} + 3 a^{2} + 6 a + 18\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 5 a^{4} + 15 a^{3} + 12 a^{2} + 2 a + 3 + \left(2 a^{4} + 2 a^{3} + 5 a^{2} + 12 a + 6\right)\cdot 19 + \left(14 a^{4} + 9 a^{3} + 5 a^{2} + 7 a + 5\right)\cdot 19^{2} + \left(8 a^{4} + 18 a^{3} + 9 a^{2} + 2 a + 3\right)\cdot 19^{3} + \left(13 a^{4} + 15 a^{3} + 17 a^{2} + 15 a + 12\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 6 a^{4} + 7 a^{3} + 10 a^{2} + 16 a + 7 + \left(9 a^{4} + 11 a^{3} + 11 a^{2} + 15 a + 15\right)\cdot 19 + \left(11 a^{4} + 6 a^{3} + 16 a^{2} + 13 a + 13\right)\cdot 19^{2} + \left(10 a^{4} + 12 a^{3} + 6 a + 10\right)\cdot 19^{3} + \left(10 a^{4} + 3 a^{3} + 11 a^{2} + 4 a\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 10 a^{4} + 12 a^{3} + 4 a^{2} + 2 a + 9 + \left(12 a^{4} + 4 a^{3} + a^{2} + 13 a + 5\right)\cdot 19 + \left(16 a^{4} + 12 a^{3} + 14 a^{2} + 2\right)\cdot 19^{2} + \left(8 a^{4} + 14 a^{3} + 18 a + 5\right)\cdot 19^{3} + \left(13 a^{4} + 7 a^{3} + 5 a^{2} + 3 a + 2\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 11 a^{4} + 11 a^{3} + 3 a^{2} + 7 a + 8 + \left(16 a^{4} + 6 a^{3} + 11 a^{2} + 2 a + 6\right)\cdot 19 + \left(13 a^{4} + 15 a^{3} + 12 a^{2} + 11 a + 4\right)\cdot 19^{2} + \left(4 a^{4} + 14 a^{3} + 18 a^{2} + 14 a + 6\right)\cdot 19^{3} + \left(8 a^{4} + 16 a^{3} + a^{2} + 15 a + 10\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 12 a^{4} + 11 a^{3} + 13 a^{2} + 5 a + 12 + \left(13 a^{4} + 8 a^{3} + 6 a^{2} + 9 a + 13\right)\cdot 19 + \left(12 a^{4} + 6 a^{3} + 3 a^{2} + 18\right)\cdot 19^{2} + \left(2 a^{4} + 6 a^{3} + 12 a^{2} + 9 a + 16\right)\cdot 19^{3} + \left(6 a^{3} + 3 a^{2} + 15 a + 15\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 11 }$ $=$ \( 13 a^{4} + a^{3} + 5 a^{2} + 8 a + 2 + \left(16 a^{4} + 5 a^{2} + 6 a + 3\right)\cdot 19 + \left(18 a^{4} + 16 a^{3} + 9 a^{2} + 13 a + 11\right)\cdot 19^{2} + \left(6 a^{4} + 13 a^{3} + 13 a^{2} + 9 a + 16\right)\cdot 19^{3} + \left(11 a^{4} + 4 a^{3} + a^{2} + 11 a + 12\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 12 }$ $=$ \( 15 a^{4} + 10 a^{3} + 15 a^{2} + a + 18 + \left(a^{3} + 9 a^{2} + 11 a + 8\right)\cdot 19 + \left(16 a^{4} + 14 a^{2} + 8 a + 12\right)\cdot 19^{2} + \left(5 a^{3} + 17 a^{2} + 14 a + 4\right)\cdot 19^{3} + \left(7 a^{4} + 8 a^{3} + 14 a^{2} + 5 a + 11\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 13 }$ $=$ \( 15 a^{4} + 12 a^{3} + 18 a^{2} + 12 a + 10 + \left(17 a^{4} + 5 a^{3} + 6 a^{2} + 7\right)\cdot 19 + \left(12 a^{4} + 13 a^{3} + 8 a^{2} + 11 a + 6\right)\cdot 19^{2} + \left(12 a^{4} + 5 a^{3} + 9 a^{2} + 7 a + 1\right)\cdot 19^{3} + \left(6 a^{4} + 8 a^{3} + 17 a^{2} + 8 a + 13\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 14 }$ $=$ \( 15 a^{4} + 16 a^{3} + 11 a^{2} + 18 a + 18 + \left(10 a^{4} + 12 a^{3} + a^{2} + 5 a + 10\right)\cdot 19 + \left(11 a^{4} + 7 a^{3} + 2 a^{2} + 12 a + 13\right)\cdot 19^{2} + \left(a^{4} + 12 a^{3} + 13 a^{2} + 15 a + 7\right)\cdot 19^{3} + \left(3 a^{4} + 5 a^{3} + 15 a^{2} + 13 a + 14\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 15 }$ $=$ \( 18 a^{4} + 9 a^{3} + 8 a^{2} + 14 a + 3 + \left(15 a^{4} + 11 a^{3} + 17 a^{2} + 15 a\right)\cdot 19 + \left(11 a^{4} + 5 a^{3} + 7 a^{2} + a + 2\right)\cdot 19^{2} + \left(4 a^{4} + 15 a^{3} + 3 a^{2} + 15 a + 7\right)\cdot 19^{3} + \left(7 a^{4} + 13 a^{3} + 5 a^{2} + 2 a + 15\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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