# Properties

 Label 2.1175.15t2.a.c Dimension $2$ Group $D_{15}$ Conductor $1175$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{15}$ Conductor: $$1175$$$$\medspace = 5^{2} \cdot 47$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 15.1.4947491410771484375.1 Galois orbit size: $4$ Smallest permutation container: $D_{15}$ Parity: odd Determinant: 1.47.2t1.a.a Projective image: $D_{15}$ Projective stem field: 15.1.4947491410771484375.1

## Defining polynomial

 $f(x)$ $=$ $$x^{15} - 6 x^{12} + 9 x^{11} + 20 x^{10} + 57 x^{9} + 29 x^{8} + 27 x^{7} - 53 x^{6} - 44 x^{5} - 84 x^{4} - 43 x^{3} - 100 x^{2} - 75 x - 55$$  .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{5} + 10 x + 35$$

Roots:
 $r_{ 1 }$ $=$ $$3 a^{4} + 16 a^{3} + 6 a^{2} + 16 a + 24 + \left(19 a^{4} + 17 a^{3} + 2 a^{2} + 6 a + 4\right)\cdot 37 + \left(11 a^{4} + 28 a^{3} + 34 a^{2} + 18 a + 18\right)\cdot 37^{2} + \left(23 a^{4} + 27 a^{3} + 11 a^{2} + 6 a + 1\right)\cdot 37^{3} + \left(16 a^{4} + 12 a^{3} + 13 a^{2} + 30 a + 22\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 2 }$ $=$ $$5 a^{4} + 22 a^{3} + 16 a^{2} + 35 a + 3 + \left(34 a^{4} + 11 a^{3} + 23 a^{2} + 34 a + 14\right)\cdot 37 + \left(33 a^{4} + 36 a^{3} + 26 a^{2} + 27 a + 12\right)\cdot 37^{2} + \left(8 a^{4} + 9 a^{3} + 4 a^{2} + 24 a + 34\right)\cdot 37^{3} + \left(19 a^{4} + 24 a^{3} + 33 a^{2} + 13 a + 5\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 3 }$ $=$ $$7 a^{4} + 6 a^{3} + 35 a^{2} + 28 a + 19 + \left(34 a^{4} + 15 a^{3} + 8 a^{2} + 12 a + 14\right)\cdot 37 + \left(26 a^{4} + 18 a^{3} + 31 a^{2} + 12 a + 30\right)\cdot 37^{2} + \left(5 a^{3} + 3 a^{2} + 15 a + 5\right)\cdot 37^{3} + \left(23 a^{4} + 25 a^{3} + 3 a^{2} + 35 a + 36\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 4 }$ $=$ $$7 a^{4} + 13 a^{3} + 28 a^{2} + 18 a + 19 + \left(33 a^{4} + 36 a^{3} + 18 a^{2} + 24 a + 6\right)\cdot 37 + \left(9 a^{4} + 5 a^{3} + 22 a^{2} + a + 5\right)\cdot 37^{2} + \left(13 a^{4} + 27 a^{3} + 10 a^{2} + 19 a + 32\right)\cdot 37^{3} + \left(21 a^{4} + 9 a^{3} + 14 a^{2} + 33 a + 22\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 5 }$ $=$ $$7 a^{4} + 27 a^{3} + 14 a^{2} + 22 a + 19 + \left(34 a^{4} + 29 a^{3} + 16 a^{2} + 9 a + 14\right)\cdot 37 + \left(22 a^{4} + 17 a^{3} + 28 a^{2} + 9 a + 35\right)\cdot 37^{2} + \left(16 a^{4} + 12 a^{3} + 6 a^{2} + 14 a + 21\right)\cdot 37^{3} + \left(5 a^{4} + 24 a^{3} + 26 a^{2} + 15 a + 6\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 6 }$ $=$ $$10 a^{4} + 33 a^{2} + 22 a + 6 + \left(23 a^{4} + 10 a^{3} + 2 a^{2} + 18 a + 1\right)\cdot 37 + \left(33 a^{4} + 33 a^{3} + 18 a^{2} + 6 a + 10\right)\cdot 37^{2} + \left(35 a^{4} + 26 a^{3} + 36 a^{2} + 13 a + 28\right)\cdot 37^{3} + \left(9 a^{4} + 3 a^{3} + 32 a^{2} + 32 a + 5\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 7 }$ $=$ $$11 a^{4} + 15 a^{3} + 3 a^{2} + 11 a + 14 + \left(24 a^{4} + 35 a^{3} + 5 a^{2} + 5 a + 9\right)\cdot 37 + \left(23 a^{4} + 32 a^{3} + 12 a^{2} + 19 a + 4\right)\cdot 37^{2} + \left(7 a^{4} + 31 a^{3} + 31 a^{2} + 18 a + 24\right)\cdot 37^{3} + \left(6 a^{4} + a^{3} + 32 a^{2} + 3 a + 12\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 8 }$ $=$ $$13 a^{4} + 21 a^{3} + 34 a^{2} + 10 a + 30 + \left(3 a^{4} + 7 a^{3} + 15 a^{2} + 28 a + 26\right)\cdot 37 + \left(29 a^{3} + 13 a^{2} + 6 a\right)\cdot 37^{2} + \left(27 a^{4} + 4 a^{3} + 20 a^{2} + 9 a + 31\right)\cdot 37^{3} + \left(34 a^{4} + 24 a^{3} + 8 a^{2} + 28 a + 18\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 9 }$ $=$ $$15 a^{4} + 9 a^{3} + 36 a^{2} + 2 a + 9 + \left(29 a^{4} + 7 a^{3} + 35 a^{2} + 11 a + 13\right)\cdot 37 + \left(17 a^{4} + 17 a^{3} + 36 a^{2} + 22 a + 31\right)\cdot 37^{2} + \left(a^{4} + 4 a^{3} + 23 a^{2} + 5 a + 11\right)\cdot 37^{3} + \left(6 a^{4} + 27 a^{3} + 19 a^{2} + a + 11\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 10 }$ $=$ $$16 a^{4} + 28 a^{3} + 20 a^{2} + 10 a + 17 + \left(10 a^{4} + 22 a^{3} + 35 a^{2} + 36 a + 9\right)\cdot 37 + \left(6 a^{4} + 2 a^{3} + 4 a^{2} + 9 a + 13\right)\cdot 37^{2} + \left(6 a^{4} + 19 a^{3} + 10 a^{2} + 12\right)\cdot 37^{3} + \left(28 a^{4} + 22 a^{3} + 16 a^{2} + 10 a + 3\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 11 }$ $=$ $$19 a^{4} + 17 a^{3} + 10 a^{2} + 35 a + 4 + \left(9 a^{4} + 31 a^{3} + 6 a^{2} + 6 a + 2\right)\cdot 37 + \left(30 a^{4} + 30 a^{3} + 10 a^{2} + 14 a + 20\right)\cdot 37^{2} + \left(23 a^{4} + 22 a^{3} + 31 a^{2} + 31 a + 5\right)\cdot 37^{3} + \left(33 a^{4} + 12 a^{3} + 3 a^{2} + 13 a + 10\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 12 }$ $=$ $$21 a^{4} + 33 a^{3} + 2 a^{2} + 16 a + 20 + \left(14 a^{4} + 32 a^{3} + 2 a^{2} + 5\right)\cdot 37 + \left(7 a^{4} + 17 a^{3} + 20 a^{2} + 2 a + 22\right)\cdot 37^{2} + \left(20 a^{3} + 18 a^{2} + 21 a + 1\right)\cdot 37^{3} + \left(30 a^{4} + 31 a^{3} + 6 a^{2} + 24 a + 18\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 13 }$ $=$ $$23 a^{4} + 19 a^{3} + 6 a^{2} + 21 a + 36 + \left(5 a^{4} + 2 a^{3} + 29 a^{2} + 25 a + 7\right)\cdot 37 + \left(25 a^{4} + 29 a^{3} + 2 a^{2} + 24 a + 16\right)\cdot 37^{2} + \left(36 a^{4} + 11 a^{3} + 2 a^{2} + 27 a + 34\right)\cdot 37^{3} + \left(29 a^{4} + 5 a^{3} + 5 a^{2} + 5 a + 17\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 14 }$ $=$ $$29 a^{4} + a^{3} + 10 a^{2} + 12 a + 10 + \left(20 a^{4} + 34 a^{3} + 10 a^{2} + 3 a + 18\right)\cdot 37 + \left(32 a^{4} + 12 a^{3} + 11 a^{2} + 26 a + 1\right)\cdot 37^{2} + \left(8 a^{4} + 32 a^{3} + 26 a^{2} + 30 a + 34\right)\cdot 37^{3} + \left(30 a^{3} + 9 a^{2} + 13 a + 1\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 15 }$ $=$ $$36 a^{4} + 32 a^{3} + 6 a^{2} + a + 29 + \left(36 a^{4} + a^{3} + 9 a^{2} + 35 a + 36\right)\cdot 37 + \left(13 a^{4} + 20 a^{3} + 23 a^{2} + 20 a\right)\cdot 37^{2} + \left(11 a^{4} + a^{3} + 20 a^{2} + 21 a + 17\right)\cdot 37^{3} + \left(31 a^{4} + 3 a^{3} + 33 a^{2} + 34 a + 28\right)\cdot 37^{4} +O(37^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.