Properties

Label 4.149169.5t5.a.a
Dimension 4
Group $S_5$
Conductor $ 3 \cdot 19 \cdot 2617 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$149169= 3 \cdot 19 \cdot 2617 $
Artin number field: Splitting field of 5.5.149169.1 defined by $f= x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.149169.2t1.a.a
Projective image: $S_5$
Projective field: Galois closure of 5.5.149169.1

Galois action

Roots of defining polynomial

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