Properties

Label 1.113.8t1.1c3
Dimension 1
Group $C_8$
Conductor $ 113 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_8$
Conductor:$113 $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 49 x^{6} - 16 x^{5} + 511 x^{4} + 367 x^{3} - 1499 x^{2} - 798 x + 1372 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_8$
Parity: Even
Corresponding Dirichlet character: \(\chi_{113}(18,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 80\cdot 97 + 37\cdot 97^{2} + 72\cdot 97^{3} + 43\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 + 34\cdot 97 + 60\cdot 97^{2} + 67\cdot 97^{3} + 2\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 27 + 82\cdot 97 + 89\cdot 97^{2} + 44\cdot 97^{3} + 13\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 30 + 40\cdot 97 + 44\cdot 97^{2} + 30\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 31 + 39\cdot 97 + 24\cdot 97^{2} + 23\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 55 + 42\cdot 97 + 38\cdot 97^{2} + 11\cdot 97^{3} + 86\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 59 + 74\cdot 97 + 29\cdot 97^{2} + 84\cdot 97^{3} + 38\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 77 + 91\cdot 97 + 62\cdot 97^{2} + 83\cdot 97^{3} + 96\cdot 97^{4} +O\left(97^{ 5 }\right)$

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

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

Character values on conjugacy classes

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