Properties

Label 56.538...256.105.a.a
Dimension $56$
Group $A_8$
Conductor $5.383\times 10^{335}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(538\!\cdots\!256\)\(\medspace = 2^{172} \cdot 43^{48} \cdot 107^{48} \cdot 179^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.81803428774472904272307991671908472193024.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.81803428774472904272307991671908472193024.1

Defining polynomial

$f(x)$$=$\(x^{8} - 28 x^{6} - 112 x^{5} - 210 x^{4} - 224 x^{3} - 140 x^{2} - 48 x + 823572\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 307 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 307 }$: \(x^{2} + 306 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 303 a + 119 + \left(115 a + 212\right)\cdot 307 + \left(238 a + 19\right)\cdot 307^{2} + \left(217 a + 39\right)\cdot 307^{3} + \left(104 a + 185\right)\cdot 307^{4} + \left(56 a + 247\right)\cdot 307^{5} + \left(238 a + 213\right)\cdot 307^{6} + \left(238 a + 73\right)\cdot 307^{7} + \left(196 a + 295\right)\cdot 307^{8} + \left(4 a + 106\right)\cdot 307^{9} +O(307^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 281 a + 246 + \left(73 a + 233\right)\cdot 307 + \left(135 a + 47\right)\cdot 307^{2} + \left(43 a + 134\right)\cdot 307^{3} + \left(228 a + 141\right)\cdot 307^{4} + \left(250 a + 230\right)\cdot 307^{5} + \left(30 a + 115\right)\cdot 307^{6} + \left(262 a + 159\right)\cdot 307^{7} + \left(205 a + 62\right)\cdot 307^{8} + \left(114 a + 46\right)\cdot 307^{9} +O(307^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 152 + 240\cdot 307 + 211\cdot 307^{2} + 270\cdot 307^{3} + 79\cdot 307^{4} + 60\cdot 307^{5} + 39\cdot 307^{6} + 41\cdot 307^{7} + 24\cdot 307^{8} + 10\cdot 307^{9} +O(307^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 115 + \left(191 a + 25\right)\cdot 307 + \left(68 a + 142\right)\cdot 307^{2} + \left(89 a + 18\right)\cdot 307^{3} + \left(202 a + 72\right)\cdot 307^{4} + \left(250 a + 199\right)\cdot 307^{5} + \left(68 a + 88\right)\cdot 307^{6} + \left(68 a + 74\right)\cdot 307^{7} + \left(110 a + 253\right)\cdot 307^{8} + \left(302 a + 221\right)\cdot 307^{9} +O(307^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 150 + 68\cdot 307 + 274\cdot 307^{2} + 301\cdot 307^{3} + 212\cdot 307^{4} + 16\cdot 307^{5} + 48\cdot 307^{6} + 196\cdot 307^{7} + 2\cdot 307^{8} + 223\cdot 307^{9} +O(307^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 65 + 304\cdot 307 + 293\cdot 307^{2} + 301\cdot 307^{3} + 246\cdot 307^{4} + 51\cdot 307^{5} + 187\cdot 307^{6} + 144\cdot 307^{7} + 50\cdot 307^{8} + 60\cdot 307^{9} +O(307^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 26 a + 220 + \left(233 a + 26\right)\cdot 307 + \left(171 a + 109\right)\cdot 307^{2} + \left(263 a + 42\right)\cdot 307^{3} + \left(78 a + 19\right)\cdot 307^{4} + \left(56 a + 253\right)\cdot 307^{5} + \left(276 a + 202\right)\cdot 307^{6} + \left(44 a + 83\right)\cdot 307^{7} + \left(101 a + 6\right)\cdot 307^{8} + \left(192 a + 262\right)\cdot 307^{9} +O(307^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 161 + 116\cdot 307 + 129\cdot 307^{2} + 119\cdot 307^{3} + 270\cdot 307^{4} + 168\cdot 307^{5} + 25\cdot 307^{6} + 148\cdot 307^{7} + 226\cdot 307^{8} + 297\cdot 307^{9} +O(307^{10})\)  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.