# Properties

 Label 2.28096.10t3.a Dimension $2$ Group $D_{10}$ Conductor $28096$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{10}$ Conductor: $$28096$$$$\medspace = 2^{6} \cdot 439$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 10.2.1217048865701888.3 Galois orbit size: $2$ Smallest permutation container: $D_{10}$ Parity: odd Projective image: $D_5$ Projective field: 5.1.192721.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $$x^{5} + 9 x + 68$$
Roots:
 $r_{ 1 }$ $=$ $$5 a^{4} + 43 a^{3} + 11 a^{2} + 43 a + 4 + \left(27 a^{4} + 49 a^{3} + 37 a^{2} + 5 a + 29\right)\cdot 73 + \left(30 a^{4} + 7 a^{3} + 26 a^{2} + 50 a + 26\right)\cdot 73^{2} + \left(8 a^{4} + 40 a^{3} + 34 a + 22\right)\cdot 73^{3} + \left(33 a^{4} + 7 a^{3} + 62 a^{2} + 63 a + 63\right)\cdot 73^{4} + \left(47 a^{4} + 28 a^{3} + 28 a^{2} + 18 a + 47\right)\cdot 73^{5} + \left(18 a^{4} + 42 a^{3} + 39 a^{2} + 18 a + 33\right)\cdot 73^{6} + \left(21 a^{4} + 48 a^{3} + 54 a^{2} + 58 a + 23\right)\cdot 73^{7} + \left(36 a^{4} + 17 a^{3} + 66 a^{2} + 70 a + 18\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 2 }$ $=$ $$5 a^{4} + 43 a^{3} + 11 a^{2} + 43 a + 68 + \left(27 a^{4} + 49 a^{3} + 37 a^{2} + 5 a + 24\right)\cdot 73 + \left(30 a^{4} + 7 a^{3} + 26 a^{2} + 50 a + 31\right)\cdot 73^{2} + \left(8 a^{4} + 40 a^{3} + 34 a + 40\right)\cdot 73^{3} + \left(33 a^{4} + 7 a^{3} + 62 a^{2} + 63 a + 48\right)\cdot 73^{4} + \left(47 a^{4} + 28 a^{3} + 28 a^{2} + 18 a + 7\right)\cdot 73^{5} + \left(18 a^{4} + 42 a^{3} + 39 a^{2} + 18 a + 45\right)\cdot 73^{6} + \left(21 a^{4} + 48 a^{3} + 54 a^{2} + 58 a + 63\right)\cdot 73^{7} + \left(36 a^{4} + 17 a^{3} + 66 a^{2} + 70 a + 7\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 3 }$ $=$ $$21 a^{4} + 18 a^{3} + 34 a^{2} + 24 a + 8 + \left(7 a^{4} + 56 a^{3} + 64 a^{2} + 49 a + 43\right)\cdot 73 + \left(50 a^{4} + 71 a^{3} + 8 a^{2} + 48 a + 56\right)\cdot 73^{2} + \left(16 a^{4} + 30 a^{3} + 66 a^{2} + 5 a + 70\right)\cdot 73^{3} + \left(39 a^{4} + 66 a^{3} + 46 a^{2} + 70 a + 4\right)\cdot 73^{4} + \left(53 a^{4} + 16 a^{3} + 67 a^{2} + 67 a + 66\right)\cdot 73^{5} + \left(45 a^{4} + 18 a^{3} + 42 a^{2} + 36 a + 64\right)\cdot 73^{6} + \left(48 a^{4} + 20 a^{3} + 62 a^{2} + 70 a + 70\right)\cdot 73^{7} + \left(21 a^{4} + 58 a^{3} + 56 a^{2} + 25 a + 33\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 4 }$ $=$ $$21 a^{4} + 18 a^{3} + 34 a^{2} + 24 a + 17 + \left(7 a^{4} + 56 a^{3} + 64 a^{2} + 49 a + 47\right)\cdot 73 + \left(50 a^{4} + 71 a^{3} + 8 a^{2} + 48 a + 51\right)\cdot 73^{2} + \left(16 a^{4} + 30 a^{3} + 66 a^{2} + 5 a + 52\right)\cdot 73^{3} + \left(39 a^{4} + 66 a^{3} + 46 a^{2} + 70 a + 19\right)\cdot 73^{4} + \left(53 a^{4} + 16 a^{3} + 67 a^{2} + 67 a + 33\right)\cdot 73^{5} + \left(45 a^{4} + 18 a^{3} + 42 a^{2} + 36 a + 53\right)\cdot 73^{6} + \left(48 a^{4} + 20 a^{3} + 62 a^{2} + 70 a + 30\right)\cdot 73^{7} + \left(21 a^{4} + 58 a^{3} + 56 a^{2} + 25 a + 44\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 5 }$ $=$ $$26 a^{4} + 31 a^{3} + 38 a^{2} + 51 a + 44 + \left(11 a^{4} + 53 a^{3} + 38 a^{2} + 46 a + 28\right)\cdot 73 + \left(3 a^{4} + 8 a^{3} + 56 a^{2} + 41 a + 54\right)\cdot 73^{2} + \left(15 a^{4} + 33 a^{3} + 65 a^{2} + 39 a + 29\right)\cdot 73^{3} + \left(29 a^{4} + 29 a^{3} + 46 a^{2} + 59 a + 20\right)\cdot 73^{4} + \left(47 a^{4} + 2 a^{3} + 47 a^{2} + 55 a + 7\right)\cdot 73^{5} + \left(41 a^{4} + 16 a^{3} + 24 a^{2} + 28 a + 50\right)\cdot 73^{6} + \left(24 a^{4} + 51 a^{3} + 47 a^{2} + 40 a + 43\right)\cdot 73^{7} + \left(30 a^{4} + 16 a^{3} + 69 a^{2} + 24 a + 52\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 6 }$ $=$ $$26 a^{4} + 31 a^{3} + 38 a^{2} + 51 a + 53 + \left(11 a^{4} + 53 a^{3} + 38 a^{2} + 46 a + 32\right)\cdot 73 + \left(3 a^{4} + 8 a^{3} + 56 a^{2} + 41 a + 49\right)\cdot 73^{2} + \left(15 a^{4} + 33 a^{3} + 65 a^{2} + 39 a + 11\right)\cdot 73^{3} + \left(29 a^{4} + 29 a^{3} + 46 a^{2} + 59 a + 35\right)\cdot 73^{4} + \left(47 a^{4} + 2 a^{3} + 47 a^{2} + 55 a + 47\right)\cdot 73^{5} + \left(41 a^{4} + 16 a^{3} + 24 a^{2} + 28 a + 38\right)\cdot 73^{6} + \left(24 a^{4} + 51 a^{3} + 47 a^{2} + 40 a + 3\right)\cdot 73^{7} + \left(30 a^{4} + 16 a^{3} + 69 a^{2} + 24 a + 63\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 7 }$ $=$ $$39 a^{4} + 67 a^{3} + 64 a^{2} + 23 a + 50 + \left(64 a^{4} + 47 a^{3} + 60 a^{2} + 61 a + 2\right)\cdot 73 + \left(58 a^{4} + 68 a^{3} + 56 a^{2} + 61 a + 3\right)\cdot 73^{2} + \left(60 a^{4} + 30 a^{3} + 31 a^{2} + 72 a + 38\right)\cdot 73^{3} + \left(16 a^{4} + 42 a^{3} + 31 a^{2} + 3 a + 33\right)\cdot 73^{4} + \left(15 a^{4} + 47 a^{3} + 53 a^{2} + 8 a + 38\right)\cdot 73^{5} + \left(46 a^{4} + 50 a^{3} + 24 a^{2} + 70 a + 24\right)\cdot 73^{6} + \left(66 a^{4} + 37 a^{3} + 69 a^{2} + 58 a + 25\right)\cdot 73^{7} + \left(63 a^{4} + 53 a^{3} + 17 a^{2} + 70 a + 2\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 8 }$ $=$ $$39 a^{4} + 67 a^{3} + 64 a^{2} + 23 a + 59 + \left(64 a^{4} + 47 a^{3} + 60 a^{2} + 61 a + 6\right)\cdot 73 + \left(58 a^{4} + 68 a^{3} + 56 a^{2} + 61 a + 71\right)\cdot 73^{2} + \left(60 a^{4} + 30 a^{3} + 31 a^{2} + 72 a + 19\right)\cdot 73^{3} + \left(16 a^{4} + 42 a^{3} + 31 a^{2} + 3 a + 48\right)\cdot 73^{4} + \left(15 a^{4} + 47 a^{3} + 53 a^{2} + 8 a + 5\right)\cdot 73^{5} + \left(46 a^{4} + 50 a^{3} + 24 a^{2} + 70 a + 13\right)\cdot 73^{6} + \left(66 a^{4} + 37 a^{3} + 69 a^{2} + 58 a + 58\right)\cdot 73^{7} + \left(63 a^{4} + 53 a^{3} + 17 a^{2} + 70 a + 12\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 9 }$ $=$ $$55 a^{4} + 60 a^{3} + 72 a^{2} + 5 a + 63 + \left(35 a^{4} + 11 a^{3} + 17 a^{2} + 56 a + 72\right)\cdot 73 + \left(3 a^{4} + 62 a^{3} + 70 a^{2} + 16 a + 12\right)\cdot 73^{2} + \left(45 a^{4} + 10 a^{3} + 54 a^{2} + 66 a + 12\right)\cdot 73^{3} + \left(27 a^{4} + 31 a^{2} + 21 a + 38\right)\cdot 73^{4} + \left(55 a^{4} + 51 a^{3} + 21 a^{2} + 68 a + 35\right)\cdot 73^{5} + \left(66 a^{4} + 18 a^{3} + 14 a^{2} + 64 a + 26\right)\cdot 73^{6} + \left(57 a^{4} + 61 a^{3} + 58 a^{2} + 63 a + 6\right)\cdot 73^{7} + \left(66 a^{4} + 72 a^{3} + 7 a^{2} + 26 a + 23\right)\cdot 73^{8} +O(73^{9})$$ $r_{ 10 }$ $=$ $$55 a^{4} + 60 a^{3} + 72 a^{2} + 5 a + 72 + \left(35 a^{4} + 11 a^{3} + 17 a^{2} + 56 a + 3\right)\cdot 73 + \left(3 a^{4} + 62 a^{3} + 70 a^{2} + 16 a + 8\right)\cdot 73^{2} + \left(45 a^{4} + 10 a^{3} + 54 a^{2} + 66 a + 67\right)\cdot 73^{3} + \left(27 a^{4} + 31 a^{2} + 21 a + 52\right)\cdot 73^{4} + \left(55 a^{4} + 51 a^{3} + 21 a^{2} + 68 a + 2\right)\cdot 73^{5} + \left(66 a^{4} + 18 a^{3} + 14 a^{2} + 64 a + 15\right)\cdot 73^{6} + \left(57 a^{4} + 61 a^{3} + 58 a^{2} + 63 a + 39\right)\cdot 73^{7} + \left(66 a^{4} + 72 a^{3} + 7 a^{2} + 26 a + 33\right)\cdot 73^{8} +O(73^{9})$$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,2)(3,4)(5,6)(7,8)(9,10)$ $-2$ $-2$ $5$ $2$ $(1,10)(2,9)(3,5)(4,6)$ $0$ $0$ $5$ $2$ $(1,9)(2,10)(3,6)(4,5)(7,8)$ $0$ $0$ $2$ $5$ $(1,4,8,6,10)(2,3,7,5,9)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $2$ $5$ $(1,8,10,4,6)(2,7,9,3,5)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $2$ $10$ $(1,3,8,5,10,2,4,7,6,9)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $2$ $10$ $(1,5,4,9,8,2,6,3,10,7)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.