Properties

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

Related objects

Learn more about

Basic invariants

Dimension: $2$
Group: $D_{10}$
Conductor: \(28096\)\(\medspace = 2^{6} \cdot 439 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 10.2.1217048865701888.3
Galois orbit size: $2$
Smallest permutation container: $D_{10}$
Parity: odd
Determinant: 1.439.2t1.a.a
Projective image: $D_5$
Projective stem field: 5.1.192721.1

Defining polynomial

$f(x)$$=$\(x^{10} - 16 x^{8} - 8 x^{7} + 89 x^{6} + 10 x^{5} - 6 x^{4} + 34 x^{3} - 236 x^{2} + 52 x - 23\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \(x^{5} + 9 x + 68\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a^{4} + 43 a^{3} + 11 a^{2} + 43 a + 4 + \left(27 a^{4} + 49 a^{3} + 37 a^{2} + 5 a + 29\right)\cdot 73 + \left(30 a^{4} + 7 a^{3} + 26 a^{2} + 50 a + 26\right)\cdot 73^{2} + \left(8 a^{4} + 40 a^{3} + 34 a + 22\right)\cdot 73^{3} + \left(33 a^{4} + 7 a^{3} + 62 a^{2} + 63 a + 63\right)\cdot 73^{4} + \left(47 a^{4} + 28 a^{3} + 28 a^{2} + 18 a + 47\right)\cdot 73^{5} + \left(18 a^{4} + 42 a^{3} + 39 a^{2} + 18 a + 33\right)\cdot 73^{6} + \left(21 a^{4} + 48 a^{3} + 54 a^{2} + 58 a + 23\right)\cdot 73^{7} + \left(36 a^{4} + 17 a^{3} + 66 a^{2} + 70 a + 18\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 a^{4} + 43 a^{3} + 11 a^{2} + 43 a + 68 + \left(27 a^{4} + 49 a^{3} + 37 a^{2} + 5 a + 24\right)\cdot 73 + \left(30 a^{4} + 7 a^{3} + 26 a^{2} + 50 a + 31\right)\cdot 73^{2} + \left(8 a^{4} + 40 a^{3} + 34 a + 40\right)\cdot 73^{3} + \left(33 a^{4} + 7 a^{3} + 62 a^{2} + 63 a + 48\right)\cdot 73^{4} + \left(47 a^{4} + 28 a^{3} + 28 a^{2} + 18 a + 7\right)\cdot 73^{5} + \left(18 a^{4} + 42 a^{3} + 39 a^{2} + 18 a + 45\right)\cdot 73^{6} + \left(21 a^{4} + 48 a^{3} + 54 a^{2} + 58 a + 63\right)\cdot 73^{7} + \left(36 a^{4} + 17 a^{3} + 66 a^{2} + 70 a + 7\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 21 a^{4} + 18 a^{3} + 34 a^{2} + 24 a + 8 + \left(7 a^{4} + 56 a^{3} + 64 a^{2} + 49 a + 43\right)\cdot 73 + \left(50 a^{4} + 71 a^{3} + 8 a^{2} + 48 a + 56\right)\cdot 73^{2} + \left(16 a^{4} + 30 a^{3} + 66 a^{2} + 5 a + 70\right)\cdot 73^{3} + \left(39 a^{4} + 66 a^{3} + 46 a^{2} + 70 a + 4\right)\cdot 73^{4} + \left(53 a^{4} + 16 a^{3} + 67 a^{2} + 67 a + 66\right)\cdot 73^{5} + \left(45 a^{4} + 18 a^{3} + 42 a^{2} + 36 a + 64\right)\cdot 73^{6} + \left(48 a^{4} + 20 a^{3} + 62 a^{2} + 70 a + 70\right)\cdot 73^{7} + \left(21 a^{4} + 58 a^{3} + 56 a^{2} + 25 a + 33\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 21 a^{4} + 18 a^{3} + 34 a^{2} + 24 a + 17 + \left(7 a^{4} + 56 a^{3} + 64 a^{2} + 49 a + 47\right)\cdot 73 + \left(50 a^{4} + 71 a^{3} + 8 a^{2} + 48 a + 51\right)\cdot 73^{2} + \left(16 a^{4} + 30 a^{3} + 66 a^{2} + 5 a + 52\right)\cdot 73^{3} + \left(39 a^{4} + 66 a^{3} + 46 a^{2} + 70 a + 19\right)\cdot 73^{4} + \left(53 a^{4} + 16 a^{3} + 67 a^{2} + 67 a + 33\right)\cdot 73^{5} + \left(45 a^{4} + 18 a^{3} + 42 a^{2} + 36 a + 53\right)\cdot 73^{6} + \left(48 a^{4} + 20 a^{3} + 62 a^{2} + 70 a + 30\right)\cdot 73^{7} + \left(21 a^{4} + 58 a^{3} + 56 a^{2} + 25 a + 44\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 26 a^{4} + 31 a^{3} + 38 a^{2} + 51 a + 44 + \left(11 a^{4} + 53 a^{3} + 38 a^{2} + 46 a + 28\right)\cdot 73 + \left(3 a^{4} + 8 a^{3} + 56 a^{2} + 41 a + 54\right)\cdot 73^{2} + \left(15 a^{4} + 33 a^{3} + 65 a^{2} + 39 a + 29\right)\cdot 73^{3} + \left(29 a^{4} + 29 a^{3} + 46 a^{2} + 59 a + 20\right)\cdot 73^{4} + \left(47 a^{4} + 2 a^{3} + 47 a^{2} + 55 a + 7\right)\cdot 73^{5} + \left(41 a^{4} + 16 a^{3} + 24 a^{2} + 28 a + 50\right)\cdot 73^{6} + \left(24 a^{4} + 51 a^{3} + 47 a^{2} + 40 a + 43\right)\cdot 73^{7} + \left(30 a^{4} + 16 a^{3} + 69 a^{2} + 24 a + 52\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 26 a^{4} + 31 a^{3} + 38 a^{2} + 51 a + 53 + \left(11 a^{4} + 53 a^{3} + 38 a^{2} + 46 a + 32\right)\cdot 73 + \left(3 a^{4} + 8 a^{3} + 56 a^{2} + 41 a + 49\right)\cdot 73^{2} + \left(15 a^{4} + 33 a^{3} + 65 a^{2} + 39 a + 11\right)\cdot 73^{3} + \left(29 a^{4} + 29 a^{3} + 46 a^{2} + 59 a + 35\right)\cdot 73^{4} + \left(47 a^{4} + 2 a^{3} + 47 a^{2} + 55 a + 47\right)\cdot 73^{5} + \left(41 a^{4} + 16 a^{3} + 24 a^{2} + 28 a + 38\right)\cdot 73^{6} + \left(24 a^{4} + 51 a^{3} + 47 a^{2} + 40 a + 3\right)\cdot 73^{7} + \left(30 a^{4} + 16 a^{3} + 69 a^{2} + 24 a + 63\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 39 a^{4} + 67 a^{3} + 64 a^{2} + 23 a + 50 + \left(64 a^{4} + 47 a^{3} + 60 a^{2} + 61 a + 2\right)\cdot 73 + \left(58 a^{4} + 68 a^{3} + 56 a^{2} + 61 a + 3\right)\cdot 73^{2} + \left(60 a^{4} + 30 a^{3} + 31 a^{2} + 72 a + 38\right)\cdot 73^{3} + \left(16 a^{4} + 42 a^{3} + 31 a^{2} + 3 a + 33\right)\cdot 73^{4} + \left(15 a^{4} + 47 a^{3} + 53 a^{2} + 8 a + 38\right)\cdot 73^{5} + \left(46 a^{4} + 50 a^{3} + 24 a^{2} + 70 a + 24\right)\cdot 73^{6} + \left(66 a^{4} + 37 a^{3} + 69 a^{2} + 58 a + 25\right)\cdot 73^{7} + \left(63 a^{4} + 53 a^{3} + 17 a^{2} + 70 a + 2\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 39 a^{4} + 67 a^{3} + 64 a^{2} + 23 a + 59 + \left(64 a^{4} + 47 a^{3} + 60 a^{2} + 61 a + 6\right)\cdot 73 + \left(58 a^{4} + 68 a^{3} + 56 a^{2} + 61 a + 71\right)\cdot 73^{2} + \left(60 a^{4} + 30 a^{3} + 31 a^{2} + 72 a + 19\right)\cdot 73^{3} + \left(16 a^{4} + 42 a^{3} + 31 a^{2} + 3 a + 48\right)\cdot 73^{4} + \left(15 a^{4} + 47 a^{3} + 53 a^{2} + 8 a + 5\right)\cdot 73^{5} + \left(46 a^{4} + 50 a^{3} + 24 a^{2} + 70 a + 13\right)\cdot 73^{6} + \left(66 a^{4} + 37 a^{3} + 69 a^{2} + 58 a + 58\right)\cdot 73^{7} + \left(63 a^{4} + 53 a^{3} + 17 a^{2} + 70 a + 12\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 55 a^{4} + 60 a^{3} + 72 a^{2} + 5 a + 63 + \left(35 a^{4} + 11 a^{3} + 17 a^{2} + 56 a + 72\right)\cdot 73 + \left(3 a^{4} + 62 a^{3} + 70 a^{2} + 16 a + 12\right)\cdot 73^{2} + \left(45 a^{4} + 10 a^{3} + 54 a^{2} + 66 a + 12\right)\cdot 73^{3} + \left(27 a^{4} + 31 a^{2} + 21 a + 38\right)\cdot 73^{4} + \left(55 a^{4} + 51 a^{3} + 21 a^{2} + 68 a + 35\right)\cdot 73^{5} + \left(66 a^{4} + 18 a^{3} + 14 a^{2} + 64 a + 26\right)\cdot 73^{6} + \left(57 a^{4} + 61 a^{3} + 58 a^{2} + 63 a + 6\right)\cdot 73^{7} + \left(66 a^{4} + 72 a^{3} + 7 a^{2} + 26 a + 23\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 55 a^{4} + 60 a^{3} + 72 a^{2} + 5 a + 72 + \left(35 a^{4} + 11 a^{3} + 17 a^{2} + 56 a + 3\right)\cdot 73 + \left(3 a^{4} + 62 a^{3} + 70 a^{2} + 16 a + 8\right)\cdot 73^{2} + \left(45 a^{4} + 10 a^{3} + 54 a^{2} + 66 a + 67\right)\cdot 73^{3} + \left(27 a^{4} + 31 a^{2} + 21 a + 52\right)\cdot 73^{4} + \left(55 a^{4} + 51 a^{3} + 21 a^{2} + 68 a + 2\right)\cdot 73^{5} + \left(66 a^{4} + 18 a^{3} + 14 a^{2} + 64 a + 15\right)\cdot 73^{6} + \left(57 a^{4} + 61 a^{3} + 58 a^{2} + 63 a + 39\right)\cdot 73^{7} + \left(66 a^{4} + 72 a^{3} + 7 a^{2} + 26 a + 33\right)\cdot 73^{8} +O(73^{9})\)  Toggle raw display

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

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

The blue line marks the conjugacy class containing complex conjugation.