Properties

Label 1.137.8t1.a.c
Dimension $1$
Group $C_8$
Conductor $137$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_8$
Conductor: \(137\)
Artin number field: Galois closure of 8.0.905824306333433.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: odd
Dirichlet character: \(\chi_{137}(10,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{8} - x^{7} + 9 x^{6} - 105 x^{5} + 954 x^{4} - 3767 x^{3} + 9149 x^{2} - 12828 x + 7607 $.

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 18 + 17\cdot 73 + 64\cdot 73^{3} + 26\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 + 40\cdot 73 + 4\cdot 73^{2} + 54\cdot 73^{3} + 19\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 36 + 55\cdot 73 + 41\cdot 73^{2} + 69\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 45 + 43\cdot 73 + 20\cdot 73^{2} + 39\cdot 73^{3} + 71\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 47 + 4\cdot 73 + 24\cdot 73^{2} + 51\cdot 73^{3} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 56 + 24\cdot 73 + 2\cdot 73^{3} + 68\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 64 + 26\cdot 73 + 40\cdot 73^{2} + 48\cdot 73^{3} + 71\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 66 + 5\cdot 73 + 14\cdot 73^{2} + 36\cdot 73^{3} + 17\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.