Properties

Label 4.7333.5t5.a.a
Dimension $4$
Group $S_5$
Conductor $7333$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(7333\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.7333.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.7333.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.1.7333.1

Defining polynomial

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

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

Roots:
$r_{ 1 }$ $=$ \( 37 + 55\cdot 431 + 94\cdot 431^{2} + 109\cdot 431^{3} + 96\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 140 + 24\cdot 431 + 359\cdot 431^{2} + 374\cdot 431^{3} + 327\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 346 + 385\cdot 431 + 355\cdot 431^{2} + 262\cdot 431^{3} + 296\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 370 + 373\cdot 431 + 388\cdot 431^{2} + 363\cdot 431^{3} + 269\cdot 431^{4} +O(431^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 402 + 22\cdot 431 + 95\cdot 431^{2} + 182\cdot 431^{3} + 302\cdot 431^{4} +O(431^{5})\) 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.