Properties

Label 4.19951.5t5.a.a
Dimension $4$
Group $S_5$
Conductor $19951$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(19951\)\(\medspace = 71 \cdot 281 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.3.19951.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: odd
Determinant: 1.19951.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.3.19951.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 198 + 203\cdot 467 + 185\cdot 467^{2} + 314\cdot 467^{3} + 409\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 338 + 186\cdot 467 + 79\cdot 467^{2} + 398\cdot 467^{3} + 270\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 433 + 369\cdot 467 + 305\cdot 467^{2} + 344\cdot 467^{3} + 48\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 437 + 388\cdot 467 + 81\cdot 467^{2} + 126\cdot 467^{3} + 349\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 464 + 251\cdot 467 + 281\cdot 467^{2} + 217\cdot 467^{3} + 322\cdot 467^{4} +O(467^{5})\) 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.