Properties

Label 6.11_89_389.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 11 \cdot 89 \cdot 389 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$380831= 11 \cdot 89 \cdot 389 $
Artin number field: Splitting field of $f= x^{7} - x^{5} - x^{4} - x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.11_89_389.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 22 a + 16 + \left(10 a + 30\right)\cdot 43 + \left(38 a + 23\right)\cdot 43^{2} + 16\cdot 43^{3} + \left(3 a + 24\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 32 a + 16 + \left(7 a + 36\right)\cdot 43 + \left(24 a + 36\right)\cdot 43^{2} + \left(33 a + 16\right)\cdot 43^{3} + \left(22 a + 17\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 a + 5 + \left(35 a + 12\right)\cdot 43 + \left(18 a + 10\right)\cdot 43^{2} + \left(9 a + 26\right)\cdot 43^{3} + \left(20 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 5 + 8\cdot 43 + 14\cdot 43^{2} + 10\cdot 43^{3} +O\left(43^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 6 a + \left(4 a + 34\right)\cdot 43 + \left(25 a + 28\right)\cdot 43^{2} + \left(29 a + 37\right)\cdot 43^{3} + \left(23 a + 29\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 37 a + 6 + \left(38 a + 32\right)\cdot 43 + \left(17 a + 6\right)\cdot 43^{2} + \left(13 a + 42\right)\cdot 43^{3} + \left(19 a + 23\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 21 a + 38 + \left(32 a + 18\right)\cdot 43 + \left(4 a + 8\right)\cdot 43^{2} + \left(42 a + 22\right)\cdot 43^{3} + \left(39 a + 26\right)\cdot 43^{4} +O\left(43^{ 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

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$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.