Properties

Label 4.126032.5t5.a.a
Dimension $4$
Group $S_5$
Conductor $126032$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(126032\)\(\medspace = 2^{4} \cdot 7877 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.126032.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.7877.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.126032.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 431 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 41 + 283\cdot 431 + 194\cdot 431^{2} + 263\cdot 431^{3} + 86\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 237 + 167\cdot 431 + 161\cdot 431^{2} + 102\cdot 431^{3} + 76\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 305 + 397\cdot 431 + 274\cdot 431^{2} + 383\cdot 431^{3} + 337\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 349 + 335\cdot 431 + 238\cdot 431^{2} + 243\cdot 431^{3} + 178\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 361 + 108\cdot 431 + 423\cdot 431^{2} + 299\cdot 431^{3} + 182\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$4$
$10$$2$$(1,2)$$2$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$-1$
$20$$6$$(1,2,3)(4,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.