Properties

Label 5.21204019456.12t183.a
Dimension $5$
Group $S_6$
Conductor $21204019456$
Indicator $1$

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:\(21204019456\)\(\medspace = 2^{8} \cdot 19^{2} \cdot 479^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.4.582464.1
Galois orbit size: $1$
Smallest permutation container: 12T183
Parity: even
Projective image: $S_6$
Projective field: 6.4.582464.1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $5$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-3$
$15$ $2$ $(1,2)$ $1$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $2$
$40$ $3$ $(1,2,3)$ $-1$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$90$ $4$ $(1,2,3,4)$ $-1$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $0$
$120$ $6$ $(1,2,3)(4,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.