Properties

Label 4.6793.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 6793 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$6793 $
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: 1.6793.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 431 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 39 + 262\cdot 431 + 185\cdot 431^{2} + 249\cdot 431^{3} + 191\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 108 + 63\cdot 431 + 206\cdot 431^{2} + 48\cdot 431^{3} + 100\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 132 + 200\cdot 431 + 311\cdot 431^{2} + 170\cdot 431^{3} + 275\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 158 + 353\cdot 431 + 323\cdot 431^{2} + 179\cdot 431^{3} + 201\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 426 + 413\cdot 431 + 265\cdot 431^{2} + 213\cdot 431^{3} + 93\cdot 431^{4} +O\left(431^{ 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.