Properties

Label 56.253...696.105.a.a
Dimension $56$
Group $A_8$
Conductor $2.537\times 10^{357}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(253\!\cdots\!696\)\(\medspace = 2^{148} \cdot 11^{48} \cdot 299471^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.1339937868468943908837290142533567581866950656.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.1339937868468943908837290142533567581866950656.1

Defining polynomial

$f(x)$$=$\(x^{8} - 112 x^{6} - 896 x^{5} - 3360 x^{4} - 7168 x^{3} - 8960 x^{2} - 6144 x + 210825792\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 863 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 863 }$: \(x^{2} + 862 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 848 + 406\cdot 863 + 148\cdot 863^{2} + 731\cdot 863^{3} + 400\cdot 863^{4} + 57\cdot 863^{5} + 590\cdot 863^{6} + 38\cdot 863^{7} + 851\cdot 863^{8} + 150\cdot 863^{9} +O(863^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 702 a + 276 + \left(854 a + 219\right)\cdot 863 + \left(804 a + 814\right)\cdot 863^{2} + \left(201 a + 229\right)\cdot 863^{3} + \left(856 a + 34\right)\cdot 863^{4} + \left(281 a + 283\right)\cdot 863^{5} + \left(165 a + 92\right)\cdot 863^{6} + \left(630 a + 93\right)\cdot 863^{7} + \left(848 a + 175\right)\cdot 863^{8} + \left(696 a + 476\right)\cdot 863^{9} +O(863^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 333 + 688\cdot 863 + 160\cdot 863^{2} + 352\cdot 863^{3} + 498\cdot 863^{4} + 438\cdot 863^{5} + 845\cdot 863^{6} + 409\cdot 863^{7} + 681\cdot 863^{8} + 638\cdot 863^{9} +O(863^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 161 a + 115 + \left(8 a + 372\right)\cdot 863 + \left(58 a + 764\right)\cdot 863^{2} + \left(661 a + 489\right)\cdot 863^{3} + \left(6 a + 688\right)\cdot 863^{4} + \left(581 a + 571\right)\cdot 863^{5} + \left(697 a + 838\right)\cdot 863^{6} + \left(232 a + 557\right)\cdot 863^{7} + \left(14 a + 393\right)\cdot 863^{8} + \left(166 a + 324\right)\cdot 863^{9} +O(863^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 134 + 666\cdot 863 + 621\cdot 863^{2} + 699\cdot 863^{3} + 517\cdot 863^{4} + 152\cdot 863^{5} + 804\cdot 863^{6} + 857\cdot 863^{7} + 392\cdot 863^{8} + 819\cdot 863^{9} +O(863^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 205 + 534\cdot 863 + 276\cdot 863^{2} + 300\cdot 863^{3} + 327\cdot 863^{4} + 375\cdot 863^{5} + 120\cdot 863^{6} + 699\cdot 863^{7} + 860\cdot 863^{8} + 286\cdot 863^{9} +O(863^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 532 a + 73 + \left(541 a + 709\right)\cdot 863 + \left(617 a + 294\right)\cdot 863^{2} + \left(587 a + 339\right)\cdot 863^{3} + \left(457 a + 557\right)\cdot 863^{4} + \left(775 a + 627\right)\cdot 863^{5} + \left(666 a + 134\right)\cdot 863^{6} + \left(278 a + 160\right)\cdot 863^{7} + \left(179 a + 98\right)\cdot 863^{8} + \left(459 a + 669\right)\cdot 863^{9} +O(863^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 331 a + 605 + \left(321 a + 718\right)\cdot 863 + \left(245 a + 370\right)\cdot 863^{2} + \left(275 a + 309\right)\cdot 863^{3} + \left(405 a + 427\right)\cdot 863^{4} + \left(87 a + 82\right)\cdot 863^{5} + \left(196 a + 26\right)\cdot 863^{6} + \left(584 a + 635\right)\cdot 863^{7} + \left(683 a + 861\right)\cdot 863^{8} + \left(403 a + 85\right)\cdot 863^{9} +O(863^{10})\)  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 value
$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.