Properties

Label 5.214066161.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $214066161$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(214066161\)\(\medspace = 3^{2} \cdot 4877^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.3.14631.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.3.14631.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a + 31 + \left(36 a + 36\right)\cdot 41 + \left(33 a + 32\right)\cdot 41^{2} + \left(15 a + 6\right)\cdot 41^{3} + \left(12 a + 36\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 36 a + 5 + \left(4 a + 17\right)\cdot 41 + \left(7 a + 16\right)\cdot 41^{2} + \left(25 a + 20\right)\cdot 41^{3} + \left(28 a + 16\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 32 + 3\cdot 41 + 31\cdot 41^{2} + 22\cdot 41^{3} + 36\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 33 + 15\cdot 41 + 13\cdot 41^{2} + 29\cdot 41^{3} + 19\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 23 + 8\cdot 41 + 29\cdot 41^{2} + 2\cdot 41^{3} + 14\cdot 41^{4} +O(41^{5})\) 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$$1$$()$$5$
$10$$2$$(1,2)$$1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$-1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.