Properties

Label 20.106...769.70.a.a
Dimension $20$
Group $S_7$
Conductor $1.068\times 10^{50}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(106\!\cdots\!769\)\(\medspace = 3^{24} \cdot 7207^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.583767.1
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 7.1.583767.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: \( x^{2} + 102x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 99 a + 97 + \left(13 a + 55\right)\cdot 103 + \left(96 a + 71\right)\cdot 103^{2} + \left(61 a + 5\right)\cdot 103^{3} + \left(33 a + 30\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 22 + 38\cdot 103 + 40\cdot 103^{2} + 82\cdot 103^{3} + 72\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 25 a + 73 + \left(74 a + 74\right)\cdot 103 + \left(90 a + 41\right)\cdot 103^{2} + \left(58 a + 1\right)\cdot 103^{3} + \left(10 a + 97\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 24 a + 3 + \left(9 a + 30\right)\cdot 103 + \left(10 a + 74\right)\cdot 103^{2} + \left(45 a + 18\right)\cdot 103^{3} + \left(26 a + 90\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 79 a + 27 + \left(93 a + 15\right)\cdot 103 + \left(92 a + 75\right)\cdot 103^{2} + \left(57 a + 53\right)\cdot 103^{3} + \left(76 a + 71\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 78 a + 98 + \left(28 a + 20\right)\cdot 103 + \left(12 a + 58\right)\cdot 103^{2} + \left(44 a + 72\right)\cdot 103^{3} + \left(92 a + 48\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 4 a + 93 + \left(89 a + 73\right)\cdot 103 + \left(6 a + 50\right)\cdot 103^{2} + \left(41 a + 74\right)\cdot 103^{3} + \left(69 a + 1\right)\cdot 103^{4} +O(103^{5})\) 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.