Properties

Label 6.237751.7t7.a
Dimension $6$
Group $S_7$
Conductor $237751$
Indicator $1$

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:\(237751\)\(\medspace = 23 \cdot 10337 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 7.1.237751.1
Galois orbit size: $1$
Smallest permutation container: $S_7$
Parity: odd
Projective image: $S_7$
Projective field: Galois closure of 7.1.237751.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 30 a + 17 + \left(16 a + 20\right)\cdot 41 + \left(17 a + 22\right)\cdot 41^{2} + \left(14 a + 22\right)\cdot 41^{3} + \left(13 a + 8\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 34 a + 37 + \left(18 a + 38\right)\cdot 41 + \left(8 a + 25\right)\cdot 41^{2} + \left(17 a + 13\right)\cdot 41^{3} + \left(3 a + 29\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 35 + 31\cdot 41 + 28\cdot 41^{2} + 33\cdot 41^{3} + 6\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 a + 16 + \left(22 a + 20\right)\cdot 41 + \left(32 a + 32\right)\cdot 41^{2} + \left(23 a + 15\right)\cdot 41^{3} + \left(37 a + 22\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 + 36\cdot 41 + 29\cdot 41^{2} + 14\cdot 41^{3} + 24\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 + 16\cdot 41 + 7\cdot 41^{2} + 15\cdot 41^{3} + 38\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 11 a + 25 + \left(24 a + 40\right)\cdot 41 + \left(23 a + 16\right)\cdot 41^{2} + \left(26 a + 7\right)\cdot 41^{3} + \left(27 a + 34\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(1,2,3,4,5,6,7)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character values
$c1$
$1$ $1$ $()$ $6$
$21$ $2$ $(1,2)$ $4$
$105$ $2$ $(1,2)(3,4)(5,6)$ $0$
$105$ $2$ $(1,2)(3,4)$ $2$
$70$ $3$ $(1,2,3)$ $3$
$280$ $3$ $(1,2,3)(4,5,6)$ $0$
$210$ $4$ $(1,2,3,4)$ $2$
$630$ $4$ $(1,2,3,4)(5,6)$ $0$
$504$ $5$ $(1,2,3,4,5)$ $1$
$210$ $6$ $(1,2,3)(4,5)(6,7)$ $-1$
$420$ $6$ $(1,2,3)(4,5)$ $1$
$840$ $6$ $(1,2,3,4,5,6)$ $0$
$720$ $7$ $(1,2,3,4,5,6,7)$ $-1$
$504$ $10$ $(1,2,3,4,5)(6,7)$ $-1$
$420$ $12$ $(1,2,3,4)(5,6,7)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.