Properties

Label 20.529...249.70.a.a
Dimension $20$
Group $S_7$
Conductor $5.298\times 10^{55}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(529\!\cdots\!249\)\(\medspace = 13^{10} \cdot 29^{10} \cdot 991^{10} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.373607.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.373607.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 + 23^{2} + 22\cdot 23^{3} + 22\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 16 + \left(2 a + 4\right)\cdot 23 + \left(18 a + 4\right)\cdot 23^{2} + \left(17 a + 4\right)\cdot 23^{3} + \left(10 a + 20\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 20 a + 22 + \left(20 a + 5\right)\cdot 23 + \left(4 a + 15\right)\cdot 23^{2} + \left(5 a + 21\right)\cdot 23^{3} + 12 a\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + 15 + \left(12 a + 13\right)\cdot 23 + \left(a + 4\right)\cdot 23^{2} + \left(10 a + 13\right)\cdot 23^{3} + \left(8 a + 11\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 13 a + 12 + \left(10 a + 5\right)\cdot 23 + \left(21 a + 18\right)\cdot 23^{2} + \left(12 a + 8\right)\cdot 23^{3} + \left(14 a + 18\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 12 a + \left(21 a + 4\right)\cdot 23 + \left(13 a + 21\right)\cdot 23^{2} + \left(2 a + 3\right)\cdot 23^{3} + \left(7 a + 3\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 11 a + 1 + \left(a + 12\right)\cdot 23 + \left(9 a + 4\right)\cdot 23^{2} + \left(20 a + 18\right)\cdot 23^{3} + \left(15 a + 14\right)\cdot 23^{4} +O(23^{5})\) 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$$1$$()$$20$
$21$$2$$(1,2)$$0$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$-4$
$70$$3$$(1,2,3)$$2$
$280$$3$$(1,2,3)(4,5,6)$$2$
$210$$4$$(1,2,3,4)$$0$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$0$
$210$$6$$(1,2,3)(4,5)(6,7)$$2$
$420$$6$$(1,2,3)(4,5)$$0$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$0$
$420$$12$$(1,2,3,4)(5,6,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.