Properties

Label 3.5e2_73e2.12t33.1c1
Dimension 3
Group $A_5$
Conductor $ 5^{2} \cdot 73^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$133225= 5^{2} \cdot 73^{2} $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + x^{2} + 4 x - 5 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 401 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 11 + 270\cdot 401 + 222\cdot 401^{2} + 85\cdot 401^{3} + 317\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 92 + 174\cdot 401 + 2\cdot 401^{2} + 357\cdot 401^{3} + 342\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 181 + 304\cdot 401 + 218\cdot 401^{2} + 26\cdot 401^{3} + 231\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 191 + 11\cdot 401 + 187\cdot 401^{2} + 250\cdot 401^{3} + 93\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 329 + 41\cdot 401 + 171\cdot 401^{2} + 82\cdot 401^{3} + 218\cdot 401^{4} +O\left(401^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2,3)$
$(3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$3$
$15$$2$$(1,2)(3,4)$$-1$
$20$$3$$(1,2,3)$$0$
$12$$5$$(1,2,3,4,5)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
$12$$5$$(1,3,4,5,2)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.