Properties

Label 5.29138831401.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $29138831401$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(29138831401\)\(\medspace = 170701^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.170701.1
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 5.5.170701.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} - 6x^{3} + 4x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 7 a + 8 + \left(10 a + 3\right)\cdot 11 + \left(5 a + 3\right)\cdot 11^{2} + \left(6 a + 5\right)\cdot 11^{3} + \left(4 a + 6\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 4\cdot 11 + 6\cdot 11^{2} + 2\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 a + 9 a\cdot 11 + \left(9 a + 5\right)\cdot 11^{2} + \left(10 a + 10\right)\cdot 11^{3} + \left(5 a + 5\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 7 + \left(a + 7\right)\cdot 11 + \left(a + 1\right)\cdot 11^{2} + \left(5 a + 8\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + 3 + 6\cdot 11 + \left(5 a + 5\right)\cdot 11^{2} + \left(4 a + 3\right)\cdot 11^{3} + \left(6 a + 7\right)\cdot 11^{4} +O(11^{5})\) 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.