Properties

Label 2.399.4t3.c.a
Dimension $2$
Group $D_4$
Conductor $399$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_4$
Conductor: \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin field: Galois closure of 8.0.25344958401.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.399.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{133})\)

Defining polynomial

$f(x)$$=$ \( x^{8} + 11x^{6} + 124x^{4} - 33x^{2} + 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 49 + 116\cdot 223 + 74\cdot 223^{2} + 150\cdot 223^{3} + 169\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 53 + 111\cdot 223 + 101\cdot 223^{2} + 223^{3} + 89\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 96 + 112\cdot 223 + 157\cdot 223^{2} + 128\cdot 223^{3} + 70\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 110 + 201\cdot 223 + 112\cdot 223^{2} + 59\cdot 223^{3} + 103\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 113 + 21\cdot 223 + 110\cdot 223^{2} + 163\cdot 223^{3} + 119\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 127 + 110\cdot 223 + 65\cdot 223^{2} + 94\cdot 223^{3} + 152\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 170 + 111\cdot 223 + 121\cdot 223^{2} + 221\cdot 223^{3} + 133\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 174 + 106\cdot 223 + 148\cdot 223^{2} + 72\cdot 223^{3} + 53\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.