Properties

Label 5.72573361.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $72573361$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(72573361\)\(\medspace = 7^{2} \cdot 1217^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.3.8519.1
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 5.3.8519.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 6 a + 13 + \left(10 a + 17\right)\cdot 43 + 27\cdot 43^{2} + \left(26 a + 2\right)\cdot 43^{3} + \left(2 a + 6\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 30 a + 5 + \left(9 a + 10\right)\cdot 43 + \left(33 a + 12\right)\cdot 43^{2} + \left(22 a + 32\right)\cdot 43^{3} + \left(24 a + 11\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 4\cdot 43 + 36\cdot 43^{2} + 29\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 13 a + 35 + \left(33 a + 32\right)\cdot 43 + \left(9 a + 35\right)\cdot 43^{2} + \left(20 a + 21\right)\cdot 43^{3} + \left(18 a + 13\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 37 a + 19 + \left(32 a + 21\right)\cdot 43 + \left(42 a + 17\right)\cdot 43^{2} + \left(16 a + 28\right)\cdot 43^{3} + \left(40 a + 25\right)\cdot 43^{4} +O(43^{5})\) 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.