Properties

Label 2.111.4t3.c.a
Dimension $2$
Group $D_4$
Conductor $111$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_4$
Conductor: \(111\)\(\medspace = 3 \cdot 37 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin field: Galois closure of 8.0.151807041.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.111.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 5x^{6} + 28x^{4} + 15x^{2} + 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 16 + 101\cdot 127 + 33\cdot 127^{2} + 28\cdot 127^{3} + 49\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 41 + 37\cdot 127 + 52\cdot 127^{2} + 77\cdot 127^{3} + 123\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 58 + 59\cdot 127 + 59\cdot 127^{2} + 68\cdot 127^{3} + 88\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 61 + 20\cdot 127 + 112\cdot 127^{2} + 110\cdot 127^{3} + 116\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 66 + 106\cdot 127 + 14\cdot 127^{2} + 16\cdot 127^{3} + 10\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 69 + 67\cdot 127 + 67\cdot 127^{2} + 58\cdot 127^{3} + 38\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 86 + 89\cdot 127 + 74\cdot 127^{2} + 49\cdot 127^{3} + 3\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 111 + 25\cdot 127 + 93\cdot 127^{2} + 98\cdot 127^{3} + 77\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display

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

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

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$$2$$(1,2)(3,5)(4,6)(7,8)$$0$
$2$$2$$(1,4)(2,3)(5,8)(6,7)$$0$
$2$$4$$(1,3,8,6)(2,4,7,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.