Properties

Label 5.3e2_5e2_11e2_139e2.10t13.1c1
Dimension 5
Group $S_5$
Conductor $ 3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 139^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:$526014225= 3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 139^{2} $
Artin number field: Splitting field of $f= x^{5} + x^{3} - 2 x^{2} - 4 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

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

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$$1$$()$$5$
$10$$2$$(1,2)$$1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$-1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$1$
The blue line marks the conjugacy class containing complex conjugation.