Properties

Label 1.97.8t1.a.c
Dimension $1$
Group $C_8$
Conductor $97$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_8$
Conductor: \(97\)
Artin field: Galois closure of 8.8.80798284478113.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: even
Dirichlet character: \(\chi_{97}(64,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} - 42x^{6} + 59x^{5} + 497x^{4} - 719x^{3} - 1792x^{2} + 2295x + 193 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 5 + 30\cdot 61 + 11\cdot 61^{2} + 10\cdot 61^{3} + 56\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 + 8\cdot 61^{2} + 57\cdot 61^{3} + 26\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 3\cdot 61 + 34\cdot 61^{2} + 52\cdot 61^{3} + 10\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 19 + 4\cdot 61^{2} + 23\cdot 61^{3} + 36\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 28 + 3\cdot 61 + 25\cdot 61^{2} + 38\cdot 61^{3} + 5\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 33 + 59\cdot 61 + 24\cdot 61^{2} + 14\cdot 61^{3} + 52\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 34 + 24\cdot 61 + 17\cdot 61^{2} + 13\cdot 61^{3} + 57\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 43 + 58\cdot 61^{2} + 34\cdot 61^{3} + 59\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display

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

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

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$$4$$(1,8,7,3)(2,5,4,6)$$\zeta_{8}^{2}$
$1$$4$$(1,3,7,8)(2,6,4,5)$$-\zeta_{8}^{2}$
$1$$8$$(1,2,8,5,7,4,3,6)$$-\zeta_{8}$
$1$$8$$(1,5,3,2,7,6,8,4)$$-\zeta_{8}^{3}$
$1$$8$$(1,4,8,6,7,2,3,5)$$\zeta_{8}$
$1$$8$$(1,6,3,4,7,5,8,2)$$\zeta_{8}^{3}$

The blue line marks the conjugacy class containing complex conjugation.