Properties

Label 2.975.4t3.b.a
Dimension $2$
Group $D_4$
Conductor $975$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_4$
Conductor: \(975\)\(\medspace = 3 \cdot 5^{2} \cdot 13 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin field: Galois closure of 8.0.1445900625.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.39.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} + x^{6} + 16x^{5} - 7x^{4} - 2x^{3} + 64x^{2} + 8x + 1 \) Copy content Toggle raw display .

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

Roots:
$r_{ 1 }$ $=$ \( 43 + 13\cdot 127 + 7\cdot 127^{2} + 69\cdot 127^{3} + 38\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 55 + 25\cdot 127 + 98\cdot 127^{2} + 107\cdot 127^{3} + 81\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 80 + 105\cdot 127 + 83\cdot 127^{2} + 34\cdot 127^{3} + 14\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 102 + 85\cdot 127 + 3\cdot 127^{2} + 74\cdot 127^{3} + 97\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 119 + 20\cdot 127 + 13\cdot 127^{2} + 119\cdot 127^{3} + 91\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 120 + 74\cdot 127 + 32\cdot 127^{2} + 21\cdot 127^{3} + 11\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 121 + 7\cdot 127 + 56\cdot 127^{2} + 93\cdot 127^{3} + 16\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 123 + 46\cdot 127 + 86\cdot 127^{2} + 115\cdot 127^{3} + 28\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.