Properties

Label 3.162409.12t33.b.a
Dimension $3$
Group $A_5$
Conductor $162409$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_5$
Conductor: \(162409\)\(\medspace = 13^{2} \cdot 31^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.156075049.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: Galois closure of 5.1.156075049.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 31x^{2} + 31x + 31 \) Copy content Toggle raw display .

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

Roots:
$r_{ 1 }$ $=$ \( 57 + 167\cdot 467 + 58\cdot 467^{2} + 389\cdot 467^{3} + 407\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 272 + 204\cdot 467 + 155\cdot 467^{2} + 14\cdot 467^{3} + 376\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 308 + 20\cdot 467 + 320\cdot 467^{2} + 154\cdot 467^{3} + 164\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 318 + 174\cdot 467 + 217\cdot 467^{2} + 306\cdot 467^{3} + 317\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 446 + 366\cdot 467 + 182\cdot 467^{2} + 69\cdot 467^{3} + 135\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2,3)$
$(3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$3$
$15$$2$$(1,2)(3,4)$$-1$
$20$$3$$(1,2,3)$$0$
$12$$5$$(1,2,3,4,5)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
$12$$5$$(1,3,4,5,2)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$

The blue line marks the conjugacy class containing complex conjugation.