Properties

Label 21.842...649.84.a.a
Dimension $21$
Group $S_7$
Conductor $8.421\times 10^{52}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $21$
Group: $S_7$
Conductor: \(842\!\cdots\!649\)\(\medspace = 29^{10} \cdot 6763^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.196127.1
Galois orbit size: $1$
Smallest permutation container: 84
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.196127.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 6 + 33\cdot 41 + 25\cdot 41^{2} + 11\cdot 41^{3} + 38\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 13 a + 36 + \left(17 a + 40\right)\cdot 41 + \left(19 a + 22\right)\cdot 41^{2} + \left(3 a + 3\right)\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 27 a + \left(5 a + 19\right)\cdot 41 + \left(29 a + 2\right)\cdot 41^{2} + \left(30 a + 21\right)\cdot 41^{3} + \left(20 a + 17\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 14 a + 40 + \left(35 a + 8\right)\cdot 41 + \left(11 a + 2\right)\cdot 41^{2} + \left(10 a + 2\right)\cdot 41^{3} + \left(20 a + 8\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 24 a + 30 + \left(9 a + 29\right)\cdot 41 + \left(4 a + 21\right)\cdot 41^{2} + \left(30 a + 22\right)\cdot 41^{3} + \left(3 a + 20\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 28 a + 34 + \left(23 a + 38\right)\cdot 41 + \left(21 a + 22\right)\cdot 41^{2} + \left(37 a + 35\right)\cdot 41^{3} + \left(40 a + 16\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 17 a + 20 + \left(31 a + 34\right)\cdot 41 + \left(36 a + 24\right)\cdot 41^{2} + \left(10 a + 26\right)\cdot 41^{3} + \left(37 a + 1\right)\cdot 41^{4} +O(41^{5})\) 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$$1$$()$$21$
$21$$2$$(1,2)$$1$
$105$$2$$(1,2)(3,4)(5,6)$$-3$
$105$$2$$(1,2)(3,4)$$1$
$70$$3$$(1,2,3)$$-3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$-1$
$630$$4$$(1,2,3,4)(5,6)$$-1$
$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)$$0$
$504$$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.