Properties

Label 4.1777.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 1777 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$1777 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + x^{3} - 2 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.1777.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 8 + \left(11 a + 8\right)\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(17 a + 13\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 a + 14 + \left(11 a + 4\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(9 a + 13\right)\cdot 23^{3} + \left(5 a + 11\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 2 a + 3 + \left(16 a + 17\right)\cdot 23 + \left(12 a + 1\right)\cdot 23^{2} + \left(8 a + 3\right)\cdot 23^{3} + \left(18 a + 7\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 15 + 14\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 21 a + 7 + \left(6 a + 1\right)\cdot 23 + \left(10 a + 11\right)\cdot 23^{2} + \left(14 a + 7\right)\cdot 23^{3} + \left(4 a + 12\right)\cdot 23^{4} +O\left(23^{ 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.