Properties

Label 20.132...481.70.a.a
Dimension $20$
Group $S_7$
Conductor $1.322\times 10^{53}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(132\!\cdots\!481\)\(\medspace = 11^{10} \cdot 13^{12} \cdot 859^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.1596881.1
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 7.3.1596881.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: \( x^{2} + 97x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 15 a + 56 + \left(84 a + 50\right)\cdot 101 + \left(47 a + 98\right)\cdot 101^{2} + \left(99 a + 100\right)\cdot 101^{3} + \left(56 a + 87\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 71 + 32\cdot 101 + 101^{2} + 34\cdot 101^{3} + 13\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 34 a + 85 + \left(14 a + 43\right)\cdot 101 + \left(98 a + 28\right)\cdot 101^{2} + \left(55 a + 97\right)\cdot 101^{3} + \left(36 a + 99\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 26 + 5\cdot 101 + 88\cdot 101^{2} + 38\cdot 101^{3} + 9\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 67 a + 19 + \left(86 a + 67\right)\cdot 101 + \left(2 a + 2\right)\cdot 101^{2} + \left(45 a + 21\right)\cdot 101^{3} + \left(64 a + 89\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 33 + 34\cdot 101 + 80\cdot 101^{2} + 64\cdot 101^{3} + 89\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 86 a + 15 + \left(16 a + 69\right)\cdot 101 + \left(53 a + 3\right)\cdot 101^{2} + \left(a + 47\right)\cdot 101^{3} + \left(44 a + 14\right)\cdot 101^{4} +O(101^{5})\) 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.