Properties

Label 28.732...384.56.a.a
Dimension $28$
Group $A_8$
Conductor $7.330\times 10^{167}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $28$
Group: $A_8$
Conductor: \(732\!\cdots\!384\)\(\medspace = 2^{86} \cdot 823547^{24}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.81784359890073480322911752930604437733376.1
Galois orbit size: $1$
Smallest permutation container: 56
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: Galois closure of 8.0.81784359890073480322911752930604437733376.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823540 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 941 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 941 }$: \( x^{2} + 940x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 145 + 701\cdot 941 + 272\cdot 941^{2} + 288\cdot 941^{3} + 311\cdot 941^{4} + 585\cdot 941^{5} + 28\cdot 941^{6} + 15\cdot 941^{7} + 839\cdot 941^{8} +O(941^{9})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 440 a + 67 + \left(881 a + 158\right)\cdot 941 + \left(823 a + 515\right)\cdot 941^{2} + \left(600 a + 639\right)\cdot 941^{3} + \left(743 a + 412\right)\cdot 941^{4} + \left(452 a + 442\right)\cdot 941^{5} + \left(323 a + 388\right)\cdot 941^{6} + \left(107 a + 222\right)\cdot 941^{7} + \left(716 a + 796\right)\cdot 941^{8} +O(941^{9})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 741 a + 681 + \left(404 a + 692\right)\cdot 941 + \left(822 a + 466\right)\cdot 941^{2} + \left(889 a + 409\right)\cdot 941^{3} + \left(276 a + 610\right)\cdot 941^{4} + \left(474 a + 686\right)\cdot 941^{5} + \left(752 a + 55\right)\cdot 941^{6} + \left(201 a + 86\right)\cdot 941^{7} + \left(781 a + 801\right)\cdot 941^{8} +O(941^{9})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 200 a + 481 + \left(536 a + 356\right)\cdot 941 + \left(118 a + 884\right)\cdot 941^{2} + \left(51 a + 476\right)\cdot 941^{3} + \left(664 a + 938\right)\cdot 941^{4} + \left(466 a + 883\right)\cdot 941^{5} + \left(188 a + 333\right)\cdot 941^{6} + \left(739 a + 476\right)\cdot 941^{7} + \left(159 a + 439\right)\cdot 941^{8} +O(941^{9})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 501 a + 507 + \left(59 a + 599\right)\cdot 941 + \left(117 a + 457\right)\cdot 941^{2} + \left(340 a + 416\right)\cdot 941^{3} + \left(197 a + 555\right)\cdot 941^{4} + \left(488 a + 151\right)\cdot 941^{5} + \left(617 a + 259\right)\cdot 941^{6} + \left(833 a + 6\right)\cdot 941^{7} + \left(224 a + 464\right)\cdot 941^{8} +O(941^{9})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 880 + 6\cdot 941 + 163\cdot 941^{2} + 642\cdot 941^{3} + 516\cdot 941^{4} + 229\cdot 941^{5} + 908\cdot 941^{6} + 511\cdot 941^{7} + 863\cdot 941^{8} +O(941^{9})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 431 + 716\cdot 941 + 530\cdot 941^{2} + 401\cdot 941^{3} + 880\cdot 941^{4} + 124\cdot 941^{5} + 826\cdot 941^{6} + 279\cdot 941^{7} + 22\cdot 941^{8} +O(941^{9})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 572 + 532\cdot 941 + 473\cdot 941^{2} + 489\cdot 941^{3} + 479\cdot 941^{4} + 659\cdot 941^{5} + 22\cdot 941^{6} + 284\cdot 941^{7} + 479\cdot 941^{8} +O(941^{9})\) 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$$1$$()$$28$
$105$$2$$(1,2)(3,4)(5,6)(7,8)$$-4$
$210$$2$$(1,2)(3,4)$$4$
$112$$3$$(1,2,3)$$1$
$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$$(1,2,3,4,5)$$-2$
$1680$$6$$(1,2,3)(4,5)(6,7)$$1$
$3360$$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.