Properties

Label 4.79_89.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 79 \cdot 89 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$7031= 79 \cdot 89 $
Artin number field: Splitting field of $f= x^{5} - x^{3} - x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Odd
Determinant: 1.79_89.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots: \[ \begin{aligned} r_{ 1 } &= 57 a + 50 + \left(35 a + 28\right)\cdot 59 + \left(49 a + 10\right)\cdot 59^{2} + \left(49 a + 53\right)\cdot 59^{3} + \left(17 a + 56\right)\cdot 59^{4} +O\left(59^{ 5 }\right) \\ r_{ 2 } &= 23 + 15\cdot 59 + 46\cdot 59^{2} + 53\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right) \\ r_{ 3 } &= 21 a + 47 + \left(52 a + 46\right)\cdot 59 + \left(45 a + 21\right)\cdot 59^{2} + \left(55 a + 3\right)\cdot 59^{3} + \left(57 a + 58\right)\cdot 59^{4} +O\left(59^{ 5 }\right) \\ r_{ 4 } &= 2 a + 48 + \left(23 a + 7\right)\cdot 59 + \left(9 a + 24\right)\cdot 59^{2} + \left(9 a + 53\right)\cdot 59^{3} + \left(41 a + 24\right)\cdot 59^{4} +O\left(59^{ 5 }\right) \\ r_{ 5 } &= 38 a + 9 + \left(6 a + 19\right)\cdot 59 + \left(13 a + 15\right)\cdot 59^{2} + \left(3 a + 13\right)\cdot 59^{3} + \left(a + 1\right)\cdot 59^{4} +O\left(59^{ 5 }\right) \\ \end{aligned}\]

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.