Properties

Label 1.41.8t1.a.d
Dimension $1$
Group $C_8$
Conductor $41$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_8$
Conductor: \(41\)
Artin field: Galois closure of 8.0.194754273881.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: odd
Dirichlet character: \(\chi_{41}(38,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} + 3x^{6} - 11x^{5} + 44x^{4} + 53x^{3} + 153x^{2} + 160x + 59 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 7\cdot 59 + 43\cdot 59^{2} + 36\cdot 59^{3} + 7\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 18 + 7\cdot 59 + 53\cdot 59^{2} + 56\cdot 59^{3} + 57\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 32 + 15\cdot 59 + 17\cdot 59^{2} + 49\cdot 59^{3} + 48\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 37 + 52\cdot 59 + 17\cdot 59^{2} + 4\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 45 + 3\cdot 59 + 54\cdot 59^{2} + 8\cdot 59^{3} + 4\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 50 + 42\cdot 59 + 8\cdot 59^{2} + 27\cdot 59^{3} + 35\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 56 + 43\cdot 59 + 41\cdot 59^{2} + 38\cdot 59^{3} + 35\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 58 + 3\cdot 59 + 18\cdot 59^{3} + 42\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,5)(2,7)(3,4)(6,8)$$-1$
$1$$4$$(1,6,5,8)(2,3,7,4)$$-\zeta_{8}^{2}$
$1$$4$$(1,8,5,6)(2,4,7,3)$$\zeta_{8}^{2}$
$1$$8$$(1,7,6,4,5,2,8,3)$$-\zeta_{8}^{3}$
$1$$8$$(1,4,8,7,5,3,6,2)$$-\zeta_{8}$
$1$$8$$(1,2,6,3,5,7,8,4)$$\zeta_{8}^{3}$
$1$$8$$(1,3,8,2,5,4,6,7)$$\zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.