Properties

Label 5.5477930481596041.6t14.a.a
Dimension 5
Group $S_5$
Conductor $ 7^{3} \cdot 25183^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:$5477930481596041= 7^{3} \cdot 25183^{3} $
Artin number field: Splitting field of 5.5.176281.1 defined by $f= x^{5} - x^{4} - 5 x^{3} + 3 x^{2} + 4 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\PGL(2,5)$
Parity: Even
Determinant: 1.176281.2t1.a.a
Projective image: $S_5$
Projective field: Galois closure of 5.5.176281.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 10 a + 4 + 12\cdot 13 + \left(a + 7\right)\cdot 13^{2} + 5\cdot 13^{3} + \left(12 a + 12\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ a + 9 + \left(5 a + 7\right)\cdot 13 + \left(a + 11\right)\cdot 13^{2} + \left(7 a + 3\right)\cdot 13^{3} + \left(11 a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 + 4\cdot 13 + 3\cdot 13^{2} + 2\cdot 13^{3} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 1 + \left(12 a + 3\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(12 a + 4\right)\cdot 13^{3} + 11\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 12 a + 10 + \left(7 a + 11\right)\cdot 13 + \left(11 a + 7\right)\cdot 13^{2} + \left(5 a + 9\right)\cdot 13^{3} + \left(a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

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.