Properties

Label 3.788.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $788$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(788\)\(\medspace = 2^{2} \cdot 197 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.788.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.197.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.0.788.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 15 + 43\cdot 59 + 55\cdot 59^{2} + 14\cdot 59^{3} + 52\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 29 + 55\cdot 59 + 3\cdot 59^{2} + 38\cdot 59^{3} + 57\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 35 + 44\cdot 59 + 12\cdot 59^{2} + 14\cdot 59^{3} + 30\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 40 + 33\cdot 59 + 45\cdot 59^{2} + 50\cdot 59^{3} + 36\cdot 59^{4} +O(59^{5})\) 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 valueComplex conjugation
$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$