Properties

Label 2.392.4t3.b.a
Dimension $2$
Group $D_4$
Conductor $392$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_4$
Conductor: \(392\)\(\medspace = 2^{3} \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin field: Galois closure of 8.0.481890304.3
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{-7})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} - 3x^{6} + 8x^{5} + x^{4} - 2x^{3} + 23x^{2} - 12x + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 2 + 11\cdot 67 + 14\cdot 67^{2} + 46\cdot 67^{3} + 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 45\cdot 67 + 12\cdot 67^{2} + 9\cdot 67^{3} + 60\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 + 53\cdot 67 + 40\cdot 67^{2} + 52\cdot 67^{3} + 52\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 25 + 60\cdot 67 + 14\cdot 67^{2} + 25\cdot 67^{3} + 18\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 29 + 58\cdot 67 + 21\cdot 67^{2} + 66\cdot 67^{3} + 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 31 + 9\cdot 67 + 64\cdot 67^{2} + 9\cdot 67^{3} + 61\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 50 + 5\cdot 67 + 33\cdot 67^{2} + 32\cdot 67^{3} + 52\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 53 + 24\cdot 67 + 66\cdot 67^{2} + 25\cdot 67^{3} + 19\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.