Galois orbit of Artin representations 2.6400.8t17.c

• Label: 2.6400.8t17.c
• Dimension: $2$
• Group: $C_4\wr C_2$
• Conductor: $6400$
• Indicator: $0$

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$6400$$\medspace = 2^{8} \cdot 5^{2}$
Artin number field:	Galois closure of 8.0.2097152000.7
Galois orbit size:	$2$
Smallest permutation container:	$C_4\wr C_2$
Parity:	odd
Projective image:	$D_4$
Projective field:	Galois closure of 4.2.8000.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 421 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$8 + 113\cdot 421 + 277\cdot 421^{2} + 130\cdot 421^{3} + 130\cdot 421^{4} + 194\cdot 421^{5} + 405\cdot 421^{6} +O(421^{7})$
$r_{ 2 }$	$=$	$138 + 164\cdot 421 + 368\cdot 421^{2} + 365\cdot 421^{3} + 85\cdot 421^{4} + 53\cdot 421^{5} + 184\cdot 421^{6} +O(421^{7})$
$r_{ 3 }$	$=$	$189 + 279\cdot 421 + 121\cdot 421^{2} + 145\cdot 421^{3} + 154\cdot 421^{5} + 9\cdot 421^{6} +O(421^{7})$
$r_{ 4 }$	$=$	$208 + 73\cdot 421 + 19\cdot 421^{2} + 285\cdot 421^{3} + 286\cdot 421^{4} + 123\cdot 421^{5} + 313\cdot 421^{6} +O(421^{7})$
$r_{ 5 }$	$=$	$213 + 347\cdot 421 + 401\cdot 421^{2} + 135\cdot 421^{3} + 134\cdot 421^{4} + 297\cdot 421^{5} + 107\cdot 421^{6} +O(421^{7})$
$r_{ 6 }$	$=$	$232 + 141\cdot 421 + 299\cdot 421^{2} + 275\cdot 421^{3} + 420\cdot 421^{4} + 266\cdot 421^{5} + 411\cdot 421^{6} +O(421^{7})$
$r_{ 7 }$	$=$	$283 + 256\cdot 421 + 52\cdot 421^{2} + 55\cdot 421^{3} + 335\cdot 421^{4} + 367\cdot 421^{5} + 236\cdot 421^{6} +O(421^{7})$
$r_{ 8 }$	$=$	$413 + 307\cdot 421 + 143\cdot 421^{2} + 290\cdot 421^{3} + 290\cdot 421^{4} + 226\cdot 421^{5} + 15\cdot 421^{6} +O(421^{7})$

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

Cycle notation

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

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

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

$(1,8)(3,6)$

$(1,3,8,6)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$1$	$2$	$(1,8)(2,7)(3,6)(4,5)$	$-2$	$-2$
$2$	$2$	$(1,8)(3,6)$	$0$	$0$
$4$	$2$	$(1,2)(3,5)(4,6)(7,8)$	$0$	$0$
$1$	$4$	$(1,6,8,3)(2,4,7,5)$	$-2 \zeta_{4}$	$2 \zeta_{4}$
$1$	$4$	$(1,3,8,6)(2,5,7,4)$	$2 \zeta_{4}$	$-2 \zeta_{4}$
$2$	$4$	$(1,3,8,6)(2,4,7,5)$	$0$	$0$
$2$	$4$	$(1,3,8,6)$	$-\zeta_{4} - 1$	$\zeta_{4} - 1$
$2$	$4$	$(1,6,8,3)$	$\zeta_{4} - 1$	$-\zeta_{4} - 1$
$2$	$4$	$(1,8)(2,4,7,5)(3,6)$	$\zeta_{4} + 1$	$-\zeta_{4} + 1$
$2$	$4$	$(1,8)(2,5,7,4)(3,6)$	$-\zeta_{4} + 1$	$\zeta_{4} + 1$
$4$	$4$	$(1,4,8,5)(2,6,7,3)$	$0$	$0$
$4$	$8$	$(1,4,6,7,8,5,3,2)$	$0$	$0$
$4$	$8$	$(1,7,3,4,8,2,6,5)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.