Properties

Label 3.90828.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $90828$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(90828\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 29^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.90828.2
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.90828.2

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 65 + 145\cdot 157 + 33\cdot 157^{2} + 155\cdot 157^{3} + 109\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 66 + 107\cdot 157 + 133\cdot 157^{2} + 40\cdot 157^{3} + 62\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 70 + 82\cdot 157 + 135\cdot 157^{2} + 40\cdot 157^{3} + 155\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 115 + 135\cdot 157 + 10\cdot 157^{2} + 77\cdot 157^{3} + 143\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.