Properties

Label 4.570025.5t4.a
Dimension $4$
Group $A_5$
Conductor $570025$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$4$
Group:$A_5$
Conductor:\(570025\)\(\medspace = 5^{2} \cdot 151^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.570025.1
Galois orbit size: $1$
Smallest permutation container: $A_5$
Parity: even
Projective image: $A_5$
Projective field: Galois closure of 5.1.570025.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 4 a + 4 + 8\cdot 11 + \left(7 a + 4\right)\cdot 11^{2} + \left(10 a + 7\right)\cdot 11^{3} + \left(6 a + 5\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 9 + \left(10 a + 5\right)\cdot 11 + \left(3 a + 10\right)\cdot 11^{2} + 9\cdot 11^{3} + 4 a\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a + 5 + \left(3 a + 9\right)\cdot 11 + \left(7 a + 2\right)\cdot 11^{2} + \left(10 a + 7\right)\cdot 11^{3} + \left(6 a + 10\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 + 10\cdot 11 + 7\cdot 11^{2} + 9\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 9 + \left(7 a + 9\right)\cdot 11 + \left(3 a + 6\right)\cdot 11^{2} + 9\cdot 11^{3} + \left(4 a + 5\right)\cdot 11^{4} +O(11^{5})\) 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 values
$c1$
$1$ $1$ $()$ $4$
$15$ $2$ $(1,2)(3,4)$ $0$
$20$ $3$ $(1,2,3)$ $1$
$12$ $5$ $(1,2,3,4,5)$ $-1$
$12$ $5$ $(1,3,4,5,2)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.