Properties

Label 56.514...416.105.a.a
Dimension $56$
Group $A_8$
Conductor $5.145\times 10^{388}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(514\!\cdots\!416\)\(\medspace = 2^{156} \cdot 67^{48} \cdot 193^{48} \cdot 1019^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.87812768645884871208063446530730170282275531390976.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.87812768645884871208063446530730170282275531390976.1

Defining polynomial

$f(x)$$=$\(x^{8} - 112 x^{6} - 896 x^{5} - 3360 x^{4} - 7168 x^{3} - 8960 x^{2} - 6144 x + 210825232\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 953 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 953 }$: \(x^{2} + 947 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 946 a + 96 + \left(128 a + 863\right)\cdot 953 + \left(350 a + 871\right)\cdot 953^{2} + \left(735 a + 214\right)\cdot 953^{3} + \left(339 a + 719\right)\cdot 953^{4} + \left(445 a + 45\right)\cdot 953^{5} + \left(421 a + 936\right)\cdot 953^{6} + \left(597 a + 295\right)\cdot 953^{7} + \left(741 a + 877\right)\cdot 953^{8} + \left(457 a + 860\right)\cdot 953^{9} +O(953^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 485 a + 426 + \left(615 a + 358\right)\cdot 953 + \left(321 a + 642\right)\cdot 953^{2} + \left(139 a + 518\right)\cdot 953^{3} + \left(389 a + 470\right)\cdot 953^{4} + \left(780 a + 564\right)\cdot 953^{5} + \left(110 a + 934\right)\cdot 953^{6} + \left(364 a + 88\right)\cdot 953^{7} + \left(132 a + 421\right)\cdot 953^{8} + \left(32 a + 868\right)\cdot 953^{9} +O(953^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 283 + 15\cdot 953 + 434\cdot 953^{2} + 82\cdot 953^{3} + 185\cdot 953^{4} + 595\cdot 953^{5} + 415\cdot 953^{6} + 145\cdot 953^{7} + 466\cdot 953^{8} + 814\cdot 953^{9} +O(953^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 7 a + 54 + \left(824 a + 691\right)\cdot 953 + \left(602 a + 937\right)\cdot 953^{2} + \left(217 a + 464\right)\cdot 953^{3} + \left(613 a + 116\right)\cdot 953^{4} + \left(507 a + 472\right)\cdot 953^{5} + \left(531 a + 160\right)\cdot 953^{6} + \left(355 a + 600\right)\cdot 953^{7} + \left(211 a + 917\right)\cdot 953^{8} + \left(495 a + 6\right)\cdot 953^{9} +O(953^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 76 + 31\cdot 953 + 811\cdot 953^{2} + 674\cdot 953^{3} + 497\cdot 953^{4} + 23\cdot 953^{5} + 76\cdot 953^{6} + 877\cdot 953^{7} + 189\cdot 953^{8} + 717\cdot 953^{9} +O(953^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 816 + 58\cdot 953 + 269\cdot 953^{2} + 651\cdot 953^{3} + 122\cdot 953^{4} + 56\cdot 953^{5} + 70\cdot 953^{6} + 220\cdot 953^{7} + 674\cdot 953^{8} + 368\cdot 953^{9} +O(953^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 468 a + 477 + \left(337 a + 707\right)\cdot 953 + \left(631 a + 50\right)\cdot 953^{2} + \left(813 a + 80\right)\cdot 953^{3} + \left(563 a + 760\right)\cdot 953^{4} + \left(172 a + 92\right)\cdot 953^{5} + \left(842 a + 819\right)\cdot 953^{6} + \left(588 a + 256\right)\cdot 953^{7} + \left(820 a + 851\right)\cdot 953^{8} + \left(920 a + 928\right)\cdot 953^{9} +O(953^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 631 + 133\cdot 953 + 748\cdot 953^{2} + 171\cdot 953^{3} + 940\cdot 953^{4} + 55\cdot 953^{5} + 400\cdot 953^{6} + 374\cdot 953^{7} + 367\cdot 953^{8} + 199\cdot 953^{9} +O(953^{10})\)  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.