Properties

Label 2.1575.8t11.a.b
Dimension $2$
Group $Q_8:C_2$
Conductor $1575$
Root number not computed
Indicator $0$

Related objects

Learn more about

Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(1575\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7 \)
Artin number field: Galois closure of 8.0.558140625.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.35.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)

Defining polynomial

$f(x)$$=$$ x^{8} - x^{7} - 10 x^{6} + 11 x^{5} + 34 x^{4} - 29 x^{3} - 55 x^{2} + 4 x + 76 $.

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 22 + 82\cdot 109 + 67\cdot 109^{2} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 28 + 48\cdot 109 + 52\cdot 109^{2} + 88\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 37 + 97\cdot 109 + 26\cdot 109^{2} + 84\cdot 109^{3} + 23\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 40 + 108\cdot 109 + 14\cdot 109^{2} + 80\cdot 109^{3} + 55\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 48 + 54\cdot 109 + 82\cdot 109^{2} + 7\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 83 + 92\cdot 109 + 99\cdot 109^{2} + 3\cdot 109^{3} + 65\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 89 + 6\cdot 109 + 6\cdot 109^{2} + 68\cdot 109^{3} + 35\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 90 + 54\cdot 109 + 85\cdot 109^{2} + 102\cdot 109^{3} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.