Properties

Label 2.119025.8t5.a.a
Dimension $2$
Group $Q_8$
Conductor $119025$
Root number $1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(119025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 23^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $1$
Artin field: Galois closure of 8.8.1686221298140625.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{69})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} - 112x^{6} + 95x^{5} + 2881x^{4} + 835x^{3} - 16858x^{2} - 25817x - 10769 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 59\cdot 89 + 83\cdot 89^{2} + 7\cdot 89^{3} + 74\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 + 17\cdot 89^{2} + 41\cdot 89^{3} + 22\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 23 + 68\cdot 89 + 66\cdot 89^{2} + 22\cdot 89^{3} + 59\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 46 + 36\cdot 89 + 72\cdot 89^{2} + 19\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 51 + 79\cdot 89 + 36\cdot 89^{2} + 81\cdot 89^{3} + 87\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 68 + 52\cdot 89 + 70\cdot 89^{2} + 52\cdot 89^{3} + 54\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 78 + 32\cdot 89 + 11\cdot 89^{2} + 52\cdot 89^{3} + 16\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 80 + 26\cdot 89 + 86\cdot 89^{2} + 7\cdot 89^{3} + 22\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,4)(5,8)(6,7)$$-2$
$2$$4$$(1,7,2,6)(3,5,4,8)$$0$
$2$$4$$(1,5,2,8)(3,6,4,7)$$0$
$2$$4$$(1,3,2,4)(5,7,8,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.