Properties

Label 21.191...632.42t418.a.a
Dimension $21$
Group $S_7$
Conductor $1.916\times 10^{39}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $21$
Group: $S_7$
Conductor: \(191\!\cdots\!632\)\(\medspace = 2^{56} \cdot 3^{47}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.725594112.1
Galois orbit size: $1$
Smallest permutation container: 42T418
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.725594112.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} + 18x^{3} - 18x^{2} - 24x - 48 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: \( x^{2} + 145x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 + 27\cdot 149 + 66\cdot 149^{2} + 34\cdot 149^{3} + 110\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a + 38 + \left(106 a + 141\right)\cdot 149 + \left(83 a + 75\right)\cdot 149^{2} + \left(3 a + 109\right)\cdot 149^{3} + \left(87 a + 31\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 141 a + 70 + \left(42 a + 110\right)\cdot 149 + \left(65 a + 6\right)\cdot 149^{2} + \left(145 a + 40\right)\cdot 149^{3} + \left(61 a + 78\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 97 a + 21 + \left(64 a + 91\right)\cdot 149 + \left(42 a + 14\right)\cdot 149^{2} + \left(30 a + 144\right)\cdot 149^{3} + \left(80 a + 148\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 52 a + 111 + \left(84 a + 103\right)\cdot 149 + \left(106 a + 119\right)\cdot 149^{2} + \left(118 a + 73\right)\cdot 149^{3} + \left(68 a + 141\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 92 + \left(21 a + 95\right)\cdot 149 + \left(59 a + 48\right)\cdot 149^{2} + \left(97 a + 6\right)\cdot 149^{3} + \left(61 a + 117\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 145 a + 108 + \left(127 a + 26\right)\cdot 149 + \left(89 a + 115\right)\cdot 149^{2} + \left(51 a + 38\right)\cdot 149^{3} + \left(87 a + 117\right)\cdot 149^{4} +O(149^{5})\) 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.