Properties

Label 3.40000.12t33.d
Dimension $3$
Group $A_5$
Conductor $40000$
Indicator $1$

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:\(40000\)\(\medspace = 2^{6} \cdot 5^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.25000000.2
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Projective image: $A_5$
Projective field: Galois closure of 5.1.25000000.2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 3 + 17 + 13\cdot 17^{2} + 17^{3} + 17^{4} + 11\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 a + 15 + \left(12 a + 12\right)\cdot 17 + \left(12 a + 15\right)\cdot 17^{2} + \left(10 a + 10\right)\cdot 17^{3} + 10 a\cdot 17^{4} + \left(4 a + 12\right)\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 a + 13 + 2 a\cdot 17 + \left(12 a + 15\right)\cdot 17^{2} + \left(8 a + 7\right)\cdot 17^{3} + \left(a + 2\right)\cdot 17^{4} + \left(a + 11\right)\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 14 a + 16 + \left(14 a + 16\right)\cdot 17 + \left(4 a + 7\right)\cdot 17^{2} + \left(8 a + 4\right)\cdot 17^{3} + \left(15 a + 12\right)\cdot 17^{4} + \left(15 a + 10\right)\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 11 a + 4 + \left(4 a + 2\right)\cdot 17 + \left(4 a + 16\right)\cdot 17^{2} + \left(6 a + 8\right)\cdot 17^{3} + 6 a\cdot 17^{4} + \left(12 a + 6\right)\cdot 17^{5} +O(17^{6})\) 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.