LMFDB - Artin representation 2.30976.8t5.b.a

Basic invariants

Dimension:	$2$
Group:	$Q_8$
Conductor:	$30976$$\medspace = 2^{8} \cdot 11^{2} $
Frobenius-Schur indicator:	$-1$
Root number:	$-1$
Artin field:	Galois closure of 8.0.29721861554176.1
Galois orbit size:	$1$
Smallest permutation container:	$Q_8$
Parity:	even
Determinant:	1.1.1t1.a.a
Projective image:	$C_2^2$
Projective field:	Galois closure of $\Q(\sqrt{2}, \sqrt{11})$

Defining polynomial

$f(x)$

$=$

$ x^{8} + 44x^{6} + 308x^{4} + 484x^{2} + 121 $

x^8 + 44*x^6 + 308*x^4 + 484*x^2 + 121

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 10.

Roots:

$r_{ 1 }$	$=$	$ 2 + 67\cdot 89 + 36\cdot 89^{2} + 87\cdot 89^{3} + 46\cdot 89^{4} + 33\cdot 89^{5} + 65\cdot 89^{6} + 57\cdot 89^{7} + 16\cdot 89^{8} + 46\cdot 89^{9} +O(89^{10})$ 2 + 6789 + 3689^2 + 8789^3 + 4689^4 + 3389^5 + 6589^6 + 5789^7 + 1689^8 + 46*89^9+O(89^10)
$r_{ 2 }$	$=$	$ 5 + 75\cdot 89 + 27\cdot 89^{2} + 73\cdot 89^{3} + 7\cdot 89^{4} + 11\cdot 89^{5} + 5\cdot 89^{6} + 3\cdot 89^{7} + 29\cdot 89^{8} + 22\cdot 89^{9} +O(89^{10})$ 5 + 7589 + 2789^2 + 7389^3 + 789^4 + 1189^5 + 589^6 + 389^7 + 2989^8 + 22*89^9+O(89^10)
$r_{ 3 }$	$=$	$ 31 + 47\cdot 89 + 8\cdot 89^{2} + 54\cdot 89^{3} + 87\cdot 89^{4} + 88\cdot 89^{5} + 20\cdot 89^{6} + 3\cdot 89^{7} + 82\cdot 89^{8} + 69\cdot 89^{9} +O(89^{10})$ 31 + 4789 + 889^2 + 5489^3 + 8789^4 + 8889^5 + 2089^6 + 389^7 + 8289^8 + 69*89^9+O(89^10)
$r_{ 4 }$	$=$	$ 37 + 80\cdot 89 + 5\cdot 89^{2} + 44\cdot 89^{3} + 10\cdot 89^{4} + 46\cdot 89^{5} + 46\cdot 89^{6} + 81\cdot 89^{7} + 52\cdot 89^{8} + 18\cdot 89^{9} +O(89^{10})$ 37 + 8089 + 589^2 + 4489^3 + 1089^4 + 4689^5 + 4689^6 + 8189^7 + 5289^8 + 18*89^9+O(89^10)
$r_{ 5 }$	$=$	$ 52 + 8\cdot 89 + 83\cdot 89^{2} + 44\cdot 89^{3} + 78\cdot 89^{4} + 42\cdot 89^{5} + 42\cdot 89^{6} + 7\cdot 89^{7} + 36\cdot 89^{8} + 70\cdot 89^{9} +O(89^{10})$ 52 + 889 + 8389^2 + 4489^3 + 7889^4 + 4289^5 + 4289^6 + 789^7 + 3689^8 + 70*89^9+O(89^10)
$r_{ 6 }$	$=$	$ 58 + 41\cdot 89 + 80\cdot 89^{2} + 34\cdot 89^{3} + 89^{4} + 68\cdot 89^{6} + 85\cdot 89^{7} + 6\cdot 89^{8} + 19\cdot 89^{9} +O(89^{10})$ 58 + 4189 + 8089^2 + 3489^3 + 89^4 + 6889^6 + 8589^7 + 689^8 + 19*89^9+O(89^10)
$r_{ 7 }$	$=$	$ 84 + 13\cdot 89 + 61\cdot 89^{2} + 15\cdot 89^{3} + 81\cdot 89^{4} + 77\cdot 89^{5} + 83\cdot 89^{6} + 85\cdot 89^{7} + 59\cdot 89^{8} + 66\cdot 89^{9} +O(89^{10})$ 84 + 1389 + 6189^2 + 1589^3 + 8189^4 + 7789^5 + 8389^6 + 8589^7 + 5989^8 + 66*89^9+O(89^10)
$r_{ 8 }$	$=$	$ 87 + 21\cdot 89 + 52\cdot 89^{2} + 89^{3} + 42\cdot 89^{4} + 55\cdot 89^{5} + 23\cdot 89^{6} + 31\cdot 89^{7} + 72\cdot 89^{8} + 42\cdot 89^{9} +O(89^{10})$ 87 + 2189 + 5289^2 + 89^3 + 4289^4 + 5589^5 + 2389^6 + 3189^7 + 7289^8 + 4289^9+O(89^10)

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,8)(2,7)(3,6)(4,5)$
$(1,2,8,7)(3,5,6,4)$
$(1,6,8,3)(2,5,7,4)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character value	Complex conjugation
$1$	$1$	$()$	$2$
$1$	$2$	$(1,8)(2,7)(3,6)(4,5)$	$-2$	✓
$2$	$4$	$(1,2,8,7)(3,5,6,4)$	$0$
$2$	$4$	$(1,6,8,3)(2,5,7,4)$	$0$
$2$	$4$	$(1,5,8,4)(2,3,7,6)$	$0$

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Basic invariants

Defining polynomial

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Character values on conjugacy classes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2.30976.8t5.b.a
Dimension	$2$
Group	$Q_8$
Conductor	$30976$
Root number	$-1$
Indicator	$-1$

$r_{ 1 }$	$=$	\( 2 + 67\cdot 89 + 36\cdot 89^{2} + 87\cdot 89^{3} + 46\cdot 89^{4} + 33\cdot 89^{5} + 65\cdot 89^{6} + 57\cdot 89^{7} + 16\cdot 89^{8} + 46\cdot 89^{9} +O(89^{10})\) 2 + 6789 + 3689^2 + 8789^3 + 4689^4 + 3389^5 + 6589^6 + 5789^7 + 1689^8 + 46*89^9+O(89^10)
$r_{ 2 }$	$=$	\( 5 + 75\cdot 89 + 27\cdot 89^{2} + 73\cdot 89^{3} + 7\cdot 89^{4} + 11\cdot 89^{5} + 5\cdot 89^{6} + 3\cdot 89^{7} + 29\cdot 89^{8} + 22\cdot 89^{9} +O(89^{10})\) 5 + 7589 + 2789^2 + 7389^3 + 789^4 + 1189^5 + 589^6 + 389^7 + 2989^8 + 22*89^9+O(89^10)
$r_{ 3 }$	$=$	\( 31 + 47\cdot 89 + 8\cdot 89^{2} + 54\cdot 89^{3} + 87\cdot 89^{4} + 88\cdot 89^{5} + 20\cdot 89^{6} + 3\cdot 89^{7} + 82\cdot 89^{8} + 69\cdot 89^{9} +O(89^{10})\) 31 + 4789 + 889^2 + 5489^3 + 8789^4 + 8889^5 + 2089^6 + 389^7 + 8289^8 + 69*89^9+O(89^10)
$r_{ 4 }$	$=$	\( 37 + 80\cdot 89 + 5\cdot 89^{2} + 44\cdot 89^{3} + 10\cdot 89^{4} + 46\cdot 89^{5} + 46\cdot 89^{6} + 81\cdot 89^{7} + 52\cdot 89^{8} + 18\cdot 89^{9} +O(89^{10})\) 37 + 8089 + 589^2 + 4489^3 + 1089^4 + 4689^5 + 4689^6 + 8189^7 + 5289^8 + 18*89^9+O(89^10)
$r_{ 5 }$	$=$	\( 52 + 8\cdot 89 + 83\cdot 89^{2} + 44\cdot 89^{3} + 78\cdot 89^{4} + 42\cdot 89^{5} + 42\cdot 89^{6} + 7\cdot 89^{7} + 36\cdot 89^{8} + 70\cdot 89^{9} +O(89^{10})\) 52 + 889 + 8389^2 + 4489^3 + 7889^4 + 4289^5 + 4289^6 + 789^7 + 3689^8 + 70*89^9+O(89^10)
$r_{ 6 }$	$=$	\( 58 + 41\cdot 89 + 80\cdot 89^{2} + 34\cdot 89^{3} + 89^{4} + 68\cdot 89^{6} + 85\cdot 89^{7} + 6\cdot 89^{8} + 19\cdot 89^{9} +O(89^{10})\) 58 + 4189 + 8089^2 + 3489^3 + 89^4 + 6889^6 + 8589^7 + 689^8 + 19*89^9+O(89^10)
$r_{ 7 }$	$=$	\( 84 + 13\cdot 89 + 61\cdot 89^{2} + 15\cdot 89^{3} + 81\cdot 89^{4} + 77\cdot 89^{5} + 83\cdot 89^{6} + 85\cdot 89^{7} + 59\cdot 89^{8} + 66\cdot 89^{9} +O(89^{10})\) 84 + 1389 + 6189^2 + 1589^3 + 8189^4 + 7789^5 + 8389^6 + 8589^7 + 5989^8 + 66*89^9+O(89^10)
$r_{ 8 }$	$=$	\( 87 + 21\cdot 89 + 52\cdot 89^{2} + 89^{3} + 42\cdot 89^{4} + 55\cdot 89^{5} + 23\cdot 89^{6} + 31\cdot 89^{7} + 72\cdot 89^{8} + 42\cdot 89^{9} +O(89^{10})\) 87 + 2189 + 5289^2 + 89^3 + 4289^4 + 5589^5 + 2389^6 + 3189^7 + 7289^8 + 4289^9+O(89^10)