Properties

Label 4.3889.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 3889 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$3889 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + x^{3} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.3889.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 15 a + 22 + \left(17 a + 28\right)\cdot 31 + \left(24 a + 7\right)\cdot 31^{2} + \left(29 a + 12\right)\cdot 31^{3} + \left(27 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 27 a + 19 + \left(20 a + 12\right)\cdot 31 + \left(2 a + 25\right)\cdot 31^{2} + \left(20 a + 19\right)\cdot 31^{3} + \left(9 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 16 a + 21 + \left(13 a + 17\right)\cdot 31 + \left(6 a + 8\right)\cdot 31^{2} + \left(a + 16\right)\cdot 31^{3} + 3 a\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a + 11 + \left(10 a + 27\right)\cdot 31 + \left(28 a + 9\right)\cdot 31^{2} + \left(10 a + 26\right)\cdot 31^{3} + \left(21 a + 24\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 21 + 6\cdot 31 + 10\cdot 31^{2} + 18\cdot 31^{3} + 5\cdot 31^{4} +O\left(31^{ 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.