Properties

Label 2.233289.8t5.b
Dimension $2$
Group $Q_8$
Conductor $233289$
Indicator $-1$

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:\(233289\)\(\medspace = 3^{2} \cdot 7^{2} \cdot 23^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin number field: Galois closure of 8.0.12696463968316569.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{21}, \sqrt{69})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 1 + 29\cdot 89^{2} + 7\cdot 89^{3} + 46\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 72\cdot 89 + 71\cdot 89^{2} + 47\cdot 89^{3} + 20\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 22 + 79\cdot 89 + 70\cdot 89^{2} + 39\cdot 89^{3} + 39\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 27 + 79\cdot 89 + 14\cdot 89^{2} + 2\cdot 89^{3} + 29\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 36 + 83\cdot 89 + 31\cdot 89^{2} + 62\cdot 89^{3} + 3\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 48 + 23\cdot 89 + 17\cdot 89^{2} + 50\cdot 89^{3} + 77\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 53 + 26\cdot 89 + 55\cdot 89^{2} + 7\cdot 89^{3} + 44\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 73 + 80\cdot 89 + 64\cdot 89^{2} + 49\cdot 89^{3} + 6\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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