Properties

Label 4.7333.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 7333 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$7333 $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 2 x^{3} - 2 x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.7333.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 431 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 37 + 55\cdot 431 + 94\cdot 431^{2} + 109\cdot 431^{3} + 96\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 140 + 24\cdot 431 + 359\cdot 431^{2} + 374\cdot 431^{3} + 327\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 346 + 385\cdot 431 + 355\cdot 431^{2} + 262\cdot 431^{3} + 296\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 370 + 373\cdot 431 + 388\cdot 431^{2} + 363\cdot 431^{3} + 269\cdot 431^{4} +O\left(431^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 402 + 22\cdot 431 + 95\cdot 431^{2} + 182\cdot 431^{3} + 302\cdot 431^{4} +O\left(431^{ 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.