Artin representation 21.19e10_56747e10.84.1c1

• Label: 21.19e10_56747e10.84.1c1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 19^{10} \cdot 56747^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$2123073742417401758726273414886165230415437366796861837601249= 19^{10} \cdot 56747^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} + 2 x^{4} - 3 x^{3} + x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	84
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 32 + 6\cdot 61 + 23\cdot 61^{2} + 56\cdot 61^{3} + 36\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 33 a + 59 + \left(53 a + 35\right)\cdot 61 + \left(49 a + 48\right)\cdot 61^{2} + \left(20 a + 38\right)\cdot 61^{3} + \left(44 a + 5\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 47 + 45\cdot 61 + 20\cdot 61^{2} + 53\cdot 61^{3} + 36\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 5 + 45\cdot 61 + 60\cdot 61^{2} + 15\cdot 61^{3} + 34\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 20 + 36\cdot 61 + 20\cdot 61^{2} + 10\cdot 61^{3} + 18\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 28 a + 31 + \left(7 a + 56\right)\cdot 61 + \left(11 a + 44\right)\cdot 61^{2} + \left(40 a + 9\right)\cdot 61^{3} + \left(16 a + 29\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 51 + 17\cdot 61 + 25\cdot 61^{2} + 59\cdot 61^{3} + 21\cdot 61^{4} +O\left(61^{ 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.