Properties

Label 45.738...368.336.a.a
Dimension $45$
Group $A_8$
Conductor $7.384\times 10^{275}$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $45$
Group: $A_8$
Conductor: \(738\!\cdots\!368\)\(\medspace = 2^{150} \cdot 823643^{39}\)
Artin stem field: Galois closure of 8.0.81841577658693500678182700989203572064256.1
Galois orbit size: $2$
Smallest permutation container: 336
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: Galois closure of 8.0.81841577658693500678182700989203572064256.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823636 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: \( x^{2} + 108x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 75 a + 29 + \left(78 a + 87\right)\cdot 109 + \left(23 a + 76\right)\cdot 109^{2} + \left(37 a + 43\right)\cdot 109^{3} + \left(54 a + 27\right)\cdot 109^{4} + \left(87 a + 99\right)\cdot 109^{5} + \left(44 a + 60\right)\cdot 109^{6} + \left(75 a + 43\right)\cdot 109^{7} + \left(60 a + 75\right)\cdot 109^{8} + \left(102 a + 34\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 108 a + 107 + \left(15 a + 80\right)\cdot 109 + \left(34 a + 15\right)\cdot 109^{2} + \left(93 a + 80\right)\cdot 109^{3} + \left(33 a + 54\right)\cdot 109^{4} + \left(105 a + 68\right)\cdot 109^{5} + \left(15 a + 65\right)\cdot 109^{6} + \left(54 a + 48\right)\cdot 109^{7} + \left(37 a + 67\right)\cdot 109^{8} + 84 a\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 48 + 13\cdot 109 + 45\cdot 109^{2} + 44\cdot 109^{3} + 42\cdot 109^{4} + 28\cdot 109^{5} + 32\cdot 109^{6} + 40\cdot 109^{7} + 24\cdot 109^{8} + 41\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 93 + 83\cdot 109^{2} + 99\cdot 109^{3} + 28\cdot 109^{4} + 97\cdot 109^{5} + 78\cdot 109^{6} + 70\cdot 109^{7} + 59\cdot 109^{8} + 85\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 99 + 13\cdot 109 + 6\cdot 109^{2} + 103\cdot 109^{3} + 97\cdot 109^{4} + 84\cdot 109^{5} + 76\cdot 109^{6} + 93\cdot 109^{7} + 94\cdot 109^{8} + 13\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 34 a + 104 + \left(30 a + 90\right)\cdot 109 + \left(85 a + 21\right)\cdot 109^{2} + \left(71 a + 57\right)\cdot 109^{3} + \left(54 a + 44\right)\cdot 109^{4} + \left(21 a + 23\right)\cdot 109^{5} + \left(64 a + 18\right)\cdot 109^{6} + \left(33 a + 74\right)\cdot 109^{7} + \left(48 a + 60\right)\cdot 109^{8} + \left(6 a + 76\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 68 + 50\cdot 109 + 44\cdot 109^{2} + 86\cdot 109^{3} + 35\cdot 109^{4} + 3\cdot 109^{5} + 18\cdot 109^{6} + 87\cdot 109^{7} + 2\cdot 109^{8} + 27\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( a + 106 + \left(93 a + 97\right)\cdot 109 + \left(74 a + 33\right)\cdot 109^{2} + \left(15 a + 30\right)\cdot 109^{3} + \left(75 a + 104\right)\cdot 109^{4} + \left(3 a + 30\right)\cdot 109^{5} + \left(93 a + 85\right)\cdot 109^{6} + \left(54 a + 86\right)\cdot 109^{7} + \left(71 a + 50\right)\cdot 109^{8} + \left(24 a + 47\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$45$
$105$$2$$(1,2)(3,4)(5,6)(7,8)$$-3$
$210$$2$$(1,2)(3,4)$$-3$
$112$$3$$(1,2,3)$$0$
$1120$$3$$(1,2,3)(4,5,6)$$0$
$1260$$4$$(1,2,3,4)(5,6,7,8)$$1$
$2520$$4$$(1,2,3,4)(5,6)$$1$
$1344$$5$$(1,2,3,4,5)$$0$
$1680$$6$$(1,2,3)(4,5)(6,7)$$0$
$3360$$6$$(1,2,3,4,5,6)(7,8)$$0$
$2880$$7$$(1,2,3,4,5,6,7)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$2880$$7$$(1,3,4,5,6,7,2)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
$1344$$15$$(1,2,3,4,5)(6,7,8)$$0$
$1344$$15$$(1,3,4,5,2)(6,7,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.