Properties

Label 3.202500.12t33.b
Dimension $3$
Group $A_5$
Conductor $202500$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:\(202500\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.5062500.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Projective image: $A_5$
Projective field: Galois closure of 5.1.5062500.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 20 + 17\cdot 31 + 3\cdot 31^{2} + 30\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 a + \left(9 a + 17\right)\cdot 31 + \left(4 a + 27\right)\cdot 31^{2} + \left(2 a + 25\right)\cdot 31^{3} + \left(12 a + 5\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 a + 19 + \left(21 a + 11\right)\cdot 31 + \left(26 a + 26\right)\cdot 31^{2} + \left(28 a + 25\right)\cdot 31^{3} + \left(18 a + 27\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 a + 16 + \left(26 a + 2\right)\cdot 31 + \left(20 a + 10\right)\cdot 31^{2} + \left(28 a + 18\right)\cdot 31^{3} + \left(9 a + 11\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 20 a + 7 + \left(4 a + 13\right)\cdot 31 + \left(10 a + 25\right)\cdot 31^{2} + \left(2 a + 23\right)\cdot 31^{3} + \left(21 a + 2\right)\cdot 31^{4} +O(31^{5})\) 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
$c1$ $c2$
$1$ $1$ $()$ $3$ $3$
$15$ $2$ $(1,2)(3,4)$ $-1$ $-1$
$20$ $3$ $(1,2,3)$ $0$ $0$
$12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.