Properties

Label 4.12753.8t35.e.a
Dimension $4$
Group $C_2 \wr C_2\wr C_2$
Conductor $12753$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_2 \wr C_2\wr C_2$
Conductor: \(12753\)\(\medspace = 3^{2} \cdot 13 \cdot 109 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.1492101.1
Galois orbit size: $1$
Smallest permutation container: $C_2 \wr C_2\wr C_2$
Parity: even
Determinant: 1.1417.2t1.a.a
Projective image: $C_2\wr C_2^2$
Projective stem field: Galois closure of 8.4.3053999169.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 823 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 39 + 731\cdot 823 + 15\cdot 823^{2} + 286\cdot 823^{3} + 156\cdot 823^{4} + 761\cdot 823^{5} +O(823^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 126 + 533\cdot 823 + 352\cdot 823^{2} + 551\cdot 823^{3} + 586\cdot 823^{4} + 88\cdot 823^{5} +O(823^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 157 + 531\cdot 823 + 205\cdot 823^{2} + 500\cdot 823^{3} + 474\cdot 823^{4} + 570\cdot 823^{5} +O(823^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 227 + 761\cdot 823 + 375\cdot 823^{2} + 408\cdot 823^{3} + 817\cdot 823^{4} + 122\cdot 823^{5} +O(823^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 338 + 423\cdot 823 + 58\cdot 823^{2} + 525\cdot 823^{3} + 692\cdot 823^{4} + 351\cdot 823^{5} +O(823^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 413 + 815\cdot 823 + 82\cdot 823^{2} + 389\cdot 823^{3} + 9\cdot 823^{4} + 626\cdot 823^{5} +O(823^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 464 + 698\cdot 823 + 420\cdot 823^{2} + 810\cdot 823^{3} + 362\cdot 823^{4} + 746\cdot 823^{5} +O(823^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 706 + 443\cdot 823 + 133\cdot 823^{2} + 644\cdot 823^{3} + 191\cdot 823^{4} + 24\cdot 823^{5} +O(823^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,3)(2,8)(4,6)(5,7)$$-4$
$2$$2$$(1,3)(5,7)$$0$
$4$$2$$(1,3)$$2$
$4$$2$$(1,7)(2,4)(3,5)(6,8)$$0$
$4$$2$$(1,3)(2,8)$$0$
$4$$2$$(1,3)(2,8)(4,6)$$-2$
$4$$2$$(1,3)(2,6)(4,8)(5,7)$$-2$
$4$$2$$(1,5)(3,7)$$2$
$8$$2$$(1,4)(2,5)(3,6)(7,8)$$0$
$8$$2$$(2,6)(4,8)(5,7)$$0$
$4$$4$$(1,7,3,5)(2,6,8,4)$$0$
$4$$4$$(1,5,3,7)$$2$
$4$$4$$(1,7,3,5)(2,8)(4,6)$$-2$
$8$$4$$(1,8,3,2)(4,5,6,7)$$0$
$8$$4$$(1,5,3,7)(2,8)$$0$
$8$$4$$(1,5)(2,6,8,4)(3,7)$$0$
$16$$4$$(1,4,7,2)(3,6,5,8)$$0$
$16$$4$$(1,8)(2,3)(4,5,6,7)$$0$
$16$$8$$(1,4,7,2,3,6,5,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.