Properties

Label 3.72075.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $72075$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(72075\)\(\medspace = 3 \cdot 5^{2} \cdot 31^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.72075.1
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 4.2.72075.1

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 10x^{2} + 10x + 15 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 11 + 58\cdot 163 + 128\cdot 163^{2} + 71\cdot 163^{3} + 49\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 46 + 34\cdot 163 + 27\cdot 163^{2} + 151\cdot 163^{3} + 138\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 127 + 69\cdot 163 + 8\cdot 163^{2} + 32\cdot 163^{3} + 52\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 143 + 162\cdot 163^{2} + 70\cdot 163^{3} + 85\cdot 163^{4} +O(163^{5})\) 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.