Properties

Label 4.11e2_37.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 11^{2} \cdot 37 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4477= 11^{2} \cdot 37 $
Artin number field: Splitting field of $f= x^{5} + x^{3} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.37.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 90 a + 50 + \left(29 a + 88\right)\cdot 101 + \left(89 a + 37\right)\cdot 101^{2} + \left(83 a + 87\right)\cdot 101^{3} + \left(36 a + 82\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 a + 6 + \left(71 a + 17\right)\cdot 101 + \left(11 a + 62\right)\cdot 101^{2} + \left(17 a + 30\right)\cdot 101^{3} + \left(64 a + 45\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 48 + 31\cdot 101 + 100\cdot 101^{2} + 32\cdot 101^{3} + 5\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 72 a + 6 + \left(42 a + 84\right)\cdot 101 + \left(46 a + 80\right)\cdot 101^{2} + \left(83 a + 83\right)\cdot 101^{3} + \left(76 a + 73\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 29 a + 92 + \left(58 a + 81\right)\cdot 101 + \left(54 a + 21\right)\cdot 101^{2} + \left(17 a + 68\right)\cdot 101^{3} + \left(24 a + 95\right)\cdot 101^{4} +O\left(101^{ 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.