Properties

Label 4.29_233.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 29 \cdot 233 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$6757= 29 \cdot 233 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 3 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.29_233.2t1.1c1

Galois action

Roots of defining polynomial

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