Properties

Label 4.2e4_461.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 2^{4} \cdot 461 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$7376= 2^{4} \cdot 461 $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 2 x^{3} - 2 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.461.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 33 a + 11 + \left(21 a + 3\right)\cdot 37 + \left(26 a + 4\right)\cdot 37^{2} + \left(36 a + 5\right)\cdot 37^{3} + \left(24 a + 33\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 16 + 34\cdot 37 + 36\cdot 37^{2} + 27\cdot 37^{3} + 31\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 35 + 32\cdot 37 + 34\cdot 37^{2} + 17\cdot 37^{3} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a + 32 + \left(15 a + 20\right)\cdot 37 + \left(10 a + 14\right)\cdot 37^{2} + 14\cdot 37^{3} + \left(12 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 19 + 19\cdot 37 + 20\cdot 37^{2} + 8\cdot 37^{3} + 23\cdot 37^{4} +O\left(37^{ 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.