Properties

Label 4.7_11_1621.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 7 \cdot 11 \cdot 1621 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$124817= 7 \cdot 11 \cdot 1621 $
Artin number field: Splitting field of $f= x^{5} - 7 x^{3} - 6 x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.7_11_1621.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 44 + 244\cdot 457 + 322\cdot 457^{2} + 104\cdot 457^{3} + 437\cdot 457^{4} +O\left(457^{ 5 }\right) \\ r_{ 2 } &= 268 + 134\cdot 457 + 199\cdot 457^{2} + 181\cdot 457^{3} + 337\cdot 457^{4} +O\left(457^{ 5 }\right) \\ r_{ 3 } &= 331 + 223\cdot 457 + 448\cdot 457^{2} + 121\cdot 457^{3} + 278\cdot 457^{4} +O\left(457^{ 5 }\right) \\ r_{ 4 } &= 361 + 161\cdot 457 + 409\cdot 457^{2} + 40\cdot 457^{3} + 119\cdot 457^{4} +O\left(457^{ 5 }\right) \\ r_{ 5 } &= 367 + 149\cdot 457 + 448\cdot 457^{2} + 7\cdot 457^{3} + 199\cdot 457^{4} +O\left(457^{ 5 }\right) \\ \end{aligned}\]

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$$1$$()$$4$
$10$$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.