Properties

Label 6.227287.7t7.a.a
Dimension $6$
Group $S_7$
Conductor $227287$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_7$
Conductor: \(227287\)\(\medspace = 167 \cdot 1361 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.227287.1
Galois orbit size: $1$
Smallest permutation container: $S_7$
Parity: odd
Determinant: 1.227287.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.227287.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 3x^{5} - 3x^{4} + 3x^{3} - 2x^{2} + 2x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 17 a + 16 + \left(17 a + 8\right)\cdot 19 + \left(11 a + 6\right)\cdot 19^{2} + \left(7 a + 18\right)\cdot 19^{3} + \left(14 a + 5\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 16\cdot 19 + 16\cdot 19^{2} + 19^{3} + 13\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 5 a + 1 + \left(3 a + 10\right)\cdot 19 + \left(17 a + 1\right)\cdot 19^{2} + \left(3 a + 15\right)\cdot 19^{3} + \left(18 a + 14\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 2 a + 14 + \left(a + 9\right)\cdot 19 + 7 a\cdot 19^{2} + \left(11 a + 14\right)\cdot 19^{3} + \left(4 a + 12\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 5 a + \left(10 a + 18\right)\cdot 19 + \left(15 a + 14\right)\cdot 19^{2} + \left(18 a + 10\right)\cdot 19^{3} + \left(6 a + 15\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 14 a + 6 + \left(15 a + 8\right)\cdot 19 + \left(a + 15\right)\cdot 19^{2} + \left(15 a + 1\right)\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 14 a + 5 + \left(8 a + 4\right)\cdot 19 + \left(3 a + 1\right)\cdot 19^{2} + 14\cdot 19^{3} + \left(12 a + 3\right)\cdot 19^{4} +O(19^{5})\) 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 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.