Properties

Label 5.10167118812786853.6t14.a.a
Dimension 5
Group $S_5$
Conductor $ 23^{3} \cdot 9419^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:$10167118812786853= 23^{3} \cdot 9419^{3} $
Artin number field: Splitting field of 5.5.216637.1 defined by $f= x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} + x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\PGL(2,5)$
Parity: Even
Determinant: 1.216637.2t1.a.a
Projective image: $S_5$
Projective field: Galois closure of 5.5.216637.1

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:
$r_{ 1 }$ $=$ $ 4 a + 5 + 3\cdot 11 + \left(10 a + 7\right)\cdot 11^{2} + \left(8 a + 10\right)\cdot 11^{3} + \left(8 a + 7\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 a + 10 + 10 a\cdot 11 + 3\cdot 11^{2} + \left(2 a + 3\right)\cdot 11^{3} + \left(2 a + 1\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 3 }$ $=$ $ a + 5 + 5\cdot 11 + \left(3 a + 8\right)\cdot 11^{2} + \left(6 a + 6\right)\cdot 11^{3} + \left(10 a + 8\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 + 7\cdot 11 + 4\cdot 11^{2} + 5\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 10 a + 9 + \left(10 a + 4\right)\cdot 11 + \left(7 a + 9\right)\cdot 11^{2} + \left(4 a + 6\right)\cdot 11^{3} +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$$1$$()$$5$
$10$$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.