Artin representation 21.3e11_8388019e11.42t418.1c1

• Label: 21.3e11_8388019e11.42t418.1c1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 3^{11} \cdot 8388019^{11}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$2562048004437783823186065825404865813155397891278422826332193592260068817290997193= 3^{11} \cdot 8388019^{11} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} - 5 x^{5} + 9 x^{4} + 7 x^{3} - 10 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T418
Parity:	Even
Determinant:	1.3_8388019.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 45 + 24\cdot 73 + 24\cdot 73^{3} + 71\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 a + 21 + \left(30 a + 7\right)\cdot 73 + \left(39 a + 18\right)\cdot 73^{2} + \left(21 a + 68\right)\cdot 73^{3} + \left(71 a + 6\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 7 + 3\cdot 73 + 10\cdot 73^{2} + 54\cdot 73^{3} + 62\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 55 a + 2 + \left(42 a + 7\right)\cdot 73 + \left(33 a + 33\right)\cdot 73^{2} + \left(51 a + 20\right)\cdot 73^{3} + \left(a + 53\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 61 a + 45 + \left(10 a + 59\right)\cdot 73 + \left(41 a + 42\right)\cdot 73^{2} + \left(52 a + 7\right)\cdot 73^{3} + \left(41 a + 41\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 12 a + 9 + \left(62 a + 31\right)\cdot 73 + \left(31 a + 9\right)\cdot 73^{2} + \left(20 a + 51\right)\cdot 73^{3} + \left(31 a + 40\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 19 + 13\cdot 73 + 32\cdot 73^{2} + 66\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

Size	Order	Action on $r_1, \ldots, r_{ 7 }$	Character value
$1$	$1$	$()$	$21$
$21$	$2$	$(1,2)$	$-1$
$105$	$2$	$(1,2)(3,4)(5,6)$	$3$
$105$	$2$	$(1,2)(3,4)$	$1$
$70$	$3$	$(1,2,3)$	$-3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.