Properties

Label 4.3_5e2_67.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 3 \cdot 5^{2} \cdot 67 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$5025= 3 \cdot 5^{2} \cdot 67 $
Artin number field: Splitting field of $f= x^{5} - x^{4} - x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.3_67.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
$r_{ 1 }$ $=$ $ 42 a + 34 + \left(31 a + 23\right)\cdot 71 + \left(13 a + 44\right)\cdot 71^{2} + \left(68 a + 53\right)\cdot 71^{3} + \left(39 a + 12\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 32 + 12\cdot 71 + 6\cdot 71^{2} + 68\cdot 71^{3} + 58\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 29 a + 47 + \left(39 a + 44\right)\cdot 71 + \left(57 a + 39\right)\cdot 71^{2} + \left(2 a + 34\right)\cdot 71^{3} + \left(31 a + 24\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 57 + 38\cdot 71^{2} + 60\cdot 71^{3} + 27\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 44 + 60\cdot 71 + 13\cdot 71^{2} + 67\cdot 71^{3} + 17\cdot 71^{4} +O\left(71^{ 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.