Properties

Label 4.67_71.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 67 \cdot 71 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4757= 67 \cdot 71 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 2 x^{3} - x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.67_71.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 4 a + 2 + \left(27 a + 11\right)\cdot 29 + \left(18 a + 9\right)\cdot 29^{2} + \left(11 a + 3\right)\cdot 29^{3} + 10\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 25 a + 22 + \left(a + 26\right)\cdot 29 + \left(10 a + 18\right)\cdot 29^{2} + \left(17 a + 13\right)\cdot 29^{3} + 28 a\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 9 + 15\cdot 29 + 4\cdot 29^{2} + 13\cdot 29^{3} + 17\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 a + 11 + \left(6 a + 26\right)\cdot 29 + \left(a + 12\right)\cdot 29^{2} + \left(10 a + 18\right)\cdot 29^{3} + \left(23 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a + 15 + \left(22 a + 7\right)\cdot 29 + \left(27 a + 12\right)\cdot 29^{2} + \left(18 a + 9\right)\cdot 29^{3} + \left(5 a + 10\right)\cdot 29^{4} +O\left(29^{ 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$$1$$()$$4$
$10$$2$$(1,2)$$2$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$-1$
$20$$6$$(1,2,3)(4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.