Properties

Label 5.515607849.6t15.b
Dimension $5$
Group $A_6$
Conductor $515607849$
Indicator $1$

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:\(515607849\)\(\medspace = 3^{6} \cdot 29^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.613089.1
Galois orbit size: $1$
Smallest permutation container: $A_6$
Parity: even
Projective image: $A_6$
Projective field: Galois closure of 6.2.613089.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 2 + 17\cdot 19 + 19^{2} + 6\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 14 + \left(a + 18\right)\cdot 19 + \left(8 a + 6\right)\cdot 19^{2} + \left(16 a + 12\right)\cdot 19^{3} + \left(5 a + 2\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 a + 1 + \left(9 a + 14\right)\cdot 19 + \left(5 a + 9\right)\cdot 19^{2} + \left(5 a + 8\right)\cdot 19^{3} + \left(9 a + 10\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 + 19^{3} + 8\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 16 a + 4 + \left(9 a + 1\right)\cdot 19 + \left(13 a + 6\right)\cdot 19^{2} + \left(13 a + 8\right)\cdot 19^{3} + \left(9 a + 14\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 10 + \left(17 a + 5\right)\cdot 19 + \left(10 a + 13\right)\cdot 19^{2} + \left(2 a + 1\right)\cdot 19^{3} + \left(13 a + 11\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2,3)$
$(1,2)(3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $5$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $2$
$40$ $3$ $(1,2,3)$ $-1$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$72$ $5$ $(1,2,3,4,5)$ $0$
$72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.