Properties

Label 2.5e2_7e2.8t7.1
Dimension 2
Group $C_8:C_2$
Conductor $ 5^{2} \cdot 7^{2}$
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:$1225= 5^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 13 x^{6} - 13 x^{5} + 25 x^{4} + 38 x^{3} - 33 x^{2} - 34 x + 11 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 281 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 59 + 20\cdot 281 + 4\cdot 281^{2} + 109\cdot 281^{3} + 26\cdot 281^{4} +O\left(281^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 94 + 154\cdot 281 + 133\cdot 281^{2} + 251\cdot 281^{3} + 160\cdot 281^{4} +O\left(281^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 125 + 163\cdot 281 + 152\cdot 281^{2} + 22\cdot 281^{3} + 108\cdot 281^{4} +O\left(281^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 142 + 113\cdot 281 + 139\cdot 281^{2} + 240\cdot 281^{3} + 199\cdot 281^{4} +O\left(281^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 151 + 253\cdot 281 + 31\cdot 281^{2} + 135\cdot 281^{3} + 244\cdot 281^{4} +O\left(281^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 152 + 105\cdot 281 + 86\cdot 281^{2} + 4\cdot 281^{3} + 177\cdot 281^{4} +O\left(281^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 172 + 245\cdot 281 + 234\cdot 281^{2} + 212\cdot 281^{3} + 125\cdot 281^{4} +O\left(281^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 230 + 67\cdot 281 + 60\cdot 281^{2} + 148\cdot 281^{3} + 81\cdot 281^{4} +O\left(281^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,8)(2,4)(3,7)(5,6)$ $-2$ $-2$
$2$ $2$ $(3,7)(5,6)$ $0$ $0$
$1$ $4$ $(1,2,8,4)(3,5,7,6)$ $2 \zeta_{4}$ $-2 \zeta_{4}$
$1$ $4$ $(1,4,8,2)(3,6,7,5)$ $-2 \zeta_{4}$ $2 \zeta_{4}$
$2$ $4$ $(1,2,8,4)(3,6,7,5)$ $0$ $0$
$2$ $8$ $(1,6,2,3,8,5,4,7)$ $0$ $0$
$2$ $8$ $(1,3,4,6,8,7,2,5)$ $0$ $0$
$2$ $8$ $(1,6,4,7,8,5,2,3)$ $0$ $0$
$2$ $8$ $(1,7,2,6,8,3,4,5)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.