Properties

Label 4.2e4_277.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 2^{4} \cdot 277 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4432= 2^{4} \cdot 277 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 2 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.277.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 12 + 13\cdot 47 + 46\cdot 47^{2} + 15\cdot 47^{3} + 18\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 45 a + 27 + \left(16 a + 7\right)\cdot 47 + \left(10 a + 18\right)\cdot 47^{2} + \left(43 a + 36\right)\cdot 47^{3} + \left(16 a + 14\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 2 a + 23 + \left(30 a + 43\right)\cdot 47 + \left(36 a + 21\right)\cdot 47^{2} + \left(3 a + 18\right)\cdot 47^{3} + \left(30 a + 5\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 a + 16 + \left(8 a + 18\right)\cdot 47 + \left(44 a + 34\right)\cdot 47^{2} + \left(39 a + 40\right)\cdot 47^{3} + \left(9 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 23 a + 17 + \left(38 a + 11\right)\cdot 47 + \left(2 a + 20\right)\cdot 47^{2} + \left(7 a + 29\right)\cdot 47^{3} + \left(37 a + 17\right)\cdot 47^{4} +O\left(47^{ 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.