Properties

Label 5.2e13_5e9_7e5.6t14.2c1
Dimension 5
Group $S_5$
Conductor $ 2^{13} \cdot 5^{9} \cdot 7^{5}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:$268912000000000= 2^{13} \cdot 5^{9} \cdot 7^{5} $
Artin number field: Splitting field of $f=x^{5} - 100 x^{3} - 500 x^{2} - 75 x + 55080$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 6T14
Parity: Even
Determinant: 1.2e3_5_7.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= -37113 +O\left(11^{ 5 }\right) \\ r_{ 2 } &= -22889 +O\left(11^{ 5 }\right) \\ r_{ 3 } &= -52917 +O\left(11^{ 5 }\right) \\ r_{ 4 } &= -52155 +O\left(11^{ 5 }\right) \\ r_{ 5 } &= 4023 +O\left(11^{ 5 }\right) \\ \end{aligned}\]

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.