Properties

Label 3.3e4_5e4.12t33.2c2
Dimension 3
Group $A_5$
Conductor $ 3^{4} \cdot 5^{4}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$50625= 3^{4} \cdot 5^{4} $
Artin number field: Splitting field of $f= x^{5} - 25 x^{2} + 75 $ 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 $\Q_{ 71 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 22 + 5\cdot 71 + 2\cdot 71^{2} + 46\cdot 71^{3} + 41\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 25 + 55\cdot 71 + 64\cdot 71^{2} + 5\cdot 71^{3} + 13\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 36 + 70\cdot 71 + 18\cdot 71^{2} + 57\cdot 71^{3} + 31\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 63 + 44\cdot 71 + 58\cdot 71^{2} + 67\cdot 71^{3} + 42\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 67 + 36\cdot 71 + 68\cdot 71^{2} + 35\cdot 71^{3} + 12\cdot 71^{4} +O\left(71^{ 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} + 1$
$12$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.