Properties

Label 3.50625.12t33.b.a
Dimension $3$
Group $A_5$
Conductor $50625$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_5$
Conductor: \(50625\)\(\medspace = 3^{4} \cdot 5^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.31640625.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: Galois closure of 5.1.31640625.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 25x^{2} + 75 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 22 + 5\cdot 71 + 2\cdot 71^{2} + 46\cdot 71^{3} + 41\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 55\cdot 71 + 64\cdot 71^{2} + 5\cdot 71^{3} + 13\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 36 + 70\cdot 71 + 18\cdot 71^{2} + 57\cdot 71^{3} + 31\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 63 + 44\cdot 71 + 58\cdot 71^{2} + 67\cdot 71^{3} + 42\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 67 + 36\cdot 71 + 68\cdot 71^{2} + 35\cdot 71^{3} + 12\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display

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.