Properties

Label 56.498...664.105.a
Dimension $56$
Group $A_8$
Conductor $4.988\times 10^{267}$
Indicator $1$

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Basic invariants

Dimension:$56$
Group:$A_8$
Conductor:\(498\!\cdots\!664\)\(\medspace = 2^{138} \cdot 51473^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.19501894337558159417628591379185664.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Projective image: $A_8$
Projective field: 8.0.19501894337558159417628591379185664.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \(x^{2} + 70 x + 5\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 59 + 21\cdot 73 + 39\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 66\cdot 73^{5} + 70\cdot 73^{6} + 68\cdot 73^{7} + 59\cdot 73^{8} + 64\cdot 73^{9} +O(73^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 48\cdot 73 + 66\cdot 73^{2} + 60\cdot 73^{3} + 34\cdot 73^{4} + 58\cdot 73^{5} + 71\cdot 73^{6} + 71\cdot 73^{7} + 31\cdot 73^{8} + 55\cdot 73^{9} +O(73^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 4 a + 33 + \left(69 a + 45\right)\cdot 73 + \left(47 a + 27\right)\cdot 73^{2} + \left(27 a + 47\right)\cdot 73^{3} + \left(36 a + 21\right)\cdot 73^{4} + \left(8 a + 47\right)\cdot 73^{5} + \left(15 a + 30\right)\cdot 73^{6} + \left(a + 59\right)\cdot 73^{7} + \left(68 a + 64\right)\cdot 73^{8} + \left(42 a + 13\right)\cdot 73^{9} +O(73^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 69 a + 45 + \left(3 a + 29\right)\cdot 73 + \left(25 a + 29\right)\cdot 73^{2} + \left(45 a + 9\right)\cdot 73^{3} + \left(36 a + 30\right)\cdot 73^{4} + \left(64 a + 36\right)\cdot 73^{5} + \left(57 a + 67\right)\cdot 73^{6} + \left(71 a + 47\right)\cdot 73^{7} + \left(4 a + 48\right)\cdot 73^{8} + \left(30 a + 1\right)\cdot 73^{9} +O(73^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 51 + 58\cdot 73 + 48\cdot 73^{2} + 11\cdot 73^{4} + 10\cdot 73^{5} + 70\cdot 73^{6} + 71\cdot 73^{7} + 22\cdot 73^{8} + 15\cdot 73^{9} +O(73^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 53 + 30\cdot 73 + 61\cdot 73^{2} + 60\cdot 73^{3} + 66\cdot 73^{4} + 73^{5} + 3\cdot 73^{6} + 38\cdot 73^{7} + 54\cdot 73^{8} +O(73^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 49 a + 60 + \left(38 a + 31\right)\cdot 73 + \left(12 a + 46\right)\cdot 73^{2} + 26 a\cdot 73^{3} + \left(7 a + 6\right)\cdot 73^{4} + \left(18 a + 12\right)\cdot 73^{5} + \left(50 a + 32\right)\cdot 73^{6} + \left(15 a + 41\right)\cdot 73^{7} + \left(56 a + 37\right)\cdot 73^{8} + \left(51 a + 20\right)\cdot 73^{9} +O(73^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 24 a + 61 + \left(34 a + 25\right)\cdot 73 + \left(60 a + 45\right)\cdot 73^{2} + \left(46 a + 66\right)\cdot 73^{3} + \left(65 a + 1\right)\cdot 73^{4} + \left(54 a + 59\right)\cdot 73^{5} + \left(22 a + 18\right)\cdot 73^{6} + \left(57 a + 38\right)\cdot 73^{7} + \left(16 a + 44\right)\cdot 73^{8} + \left(21 a + 46\right)\cdot 73^{9} +O(73^{10})\)  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 values
$c1$
$1$ $1$ $()$ $56$
$105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $8$
$210$ $2$ $(1,2)(3,4)$ $0$
$112$ $3$ $(1,2,3)$ $-4$
$1120$ $3$ $(1,2,3)(4,5,6)$ $-1$
$1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $0$
$2520$ $4$ $(1,2,3,4)(5,6)$ $0$
$1344$ $5$ $(1,2,3,4,5)$ $1$
$1680$ $6$ $(1,2,3)(4,5)(6,7)$ $0$
$3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $-1$
$2880$ $7$ $(1,2,3,4,5,6,7)$ $0$
$2880$ $7$ $(1,3,4,5,6,7,2)$ $0$
$1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $1$
$1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $1$
The blue line marks the conjugacy class containing complex conjugation.