Properties

Label 4.5e2_11e3_29.8t35.4c1
Dimension 4
Group $C_2 \wr C_2\wr C_2$
Conductor $ 5^{2} \cdot 11^{3} \cdot 29 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_2 \wr C_2\wr C_2$
Conductor:$964975= 5^{2} \cdot 11^{3} \cdot 29 $
Artin number field: Splitting field of $f=x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2 \wr C_2\wr C_2$
Parity: Odd
Determinant: 1.11_29.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 6.
Roots: \[ \begin{aligned} r_{ 1 } &= -149976192415541806 +O\left(929^{ 6 }\right) \\ r_{ 2 } &= -78517384993180892 +O\left(929^{ 6 }\right) \\ r_{ 3 } &= -179400370137705934 +O\left(929^{ 6 }\right) \\ r_{ 4 } &= -118739206366129749 +O\left(929^{ 6 }\right) \\ r_{ 5 } &= -194711489019800431 +O\left(929^{ 6 }\right) \\ r_{ 6 } &= -279769572953102002 +O\left(929^{ 6 }\right) \\ r_{ 7 } &= -308991004696435370 +O\left(929^{ 6 }\right) \\ r_{ 8 } &= 24450704703540344 +O\left(929^{ 6 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,4)(2,7)(3,5)(6,8)$$-4$
$2$$2$$(2,7)(6,8)$$0$
$4$$2$$(2,7)$$2$
$4$$2$$(1,3)(2,6)(4,5)(7,8)$$0$
$4$$2$$(2,7)(3,5)$$0$
$4$$2$$(2,6)(7,8)$$-2$
$4$$2$$(1,4)(2,7)(6,8)$$-2$
$4$$2$$(1,3)(2,7)(4,5)(6,8)$$2$
$8$$2$$(1,6)(2,3)(4,8)(5,7)$$0$
$8$$2$$(1,4)(2,6)(7,8)$$0$
$4$$4$$(1,3,4,5)(2,8,7,6)$$0$
$4$$4$$(2,8,7,6)$$-2$
$4$$4$$(1,5,4,3)(2,7)(6,8)$$2$
$8$$4$$(1,6,4,8)(2,5,7,3)$$0$
$8$$4$$(1,4)(2,8,7,6)$$0$
$8$$4$$(1,3)(2,8,7,6)(4,5)$$0$
$16$$4$$(1,6,3,2)(4,8,5,7)$$0$
$16$$4$$(1,6)(2,5,7,3)(4,8)$$0$
$16$$8$$(1,6,3,2,4,8,5,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.