Properties

Label 4.2e4_349.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 2^{4} \cdot 349 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$5584= 2^{4} \cdot 349 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.349.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 7 a + 11 + \left(2 a + 6\right)\cdot 19 + \left(13 a + 14\right)\cdot 19^{2} + \left(4 a + 1\right)\cdot 19^{3} + \left(13 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 a + 2 + \left(3 a + 9\right)\cdot 19 + \left(12 a + 7\right)\cdot 19^{2} + \left(9 a + 17\right)\cdot 19^{3} + \left(4 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 2 + 12\cdot 19 + 12\cdot 19^{2} + 10\cdot 19^{3} + 6\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 12 a + 18 + \left(16 a + 1\right)\cdot 19 + \left(5 a + 6\right)\cdot 19^{2} + \left(14 a + 12\right)\cdot 19^{3} + \left(5 a + 11\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 15 a + 6 + \left(15 a + 8\right)\cdot 19 + \left(6 a + 16\right)\cdot 19^{2} + \left(9 a + 14\right)\cdot 19^{3} + \left(14 a + 5\right)\cdot 19^{4} +O\left(19^{ 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.