Properties

Label 21.147...801.84.a.a
Dimension $21$
Group $S_7$
Conductor $1.475\times 10^{53}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $21$
Group: $S_7$
Conductor: \(147\!\cdots\!801\)\(\medspace = 7^{10} \cdot 29633^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.1452017.1
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 7.3.1452017.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{5} - 2x^{4} + 4x^{2} + x - 1 \) 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:
$r_{ 1 }$ $=$ \( 7 a + 77 + \left(56 a + 6\right)\cdot 149 + \left(66 a + 26\right)\cdot 149^{2} + \left(73 a + 28\right)\cdot 149^{3} + \left(119 a + 110\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 96 a + 10 + \left(135 a + 17\right)\cdot 149 + \left(14 a + 84\right)\cdot 149^{2} + \left(109 a + 129\right)\cdot 149^{3} + \left(84 a + 142\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 104 + 35\cdot 149 + 116\cdot 149^{2} + 61\cdot 149^{3} + 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 53 a + 96 + \left(13 a + 16\right)\cdot 149 + \left(134 a + 8\right)\cdot 149^{2} + \left(39 a + 104\right)\cdot 149^{3} + \left(64 a + 74\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 23 a + 56 + \left(89 a + 130\right)\cdot 149 + \left(59 a + 62\right)\cdot 149^{2} + \left(113 a + 109\right)\cdot 149^{3} + \left(54 a + 46\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 126 a + 148 + \left(59 a + 16\right)\cdot 149 + \left(89 a + 63\right)\cdot 149^{2} + \left(35 a + 56\right)\cdot 149^{3} + \left(94 a + 3\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 142 a + 105 + \left(92 a + 74\right)\cdot 149 + \left(82 a + 86\right)\cdot 149^{2} + \left(75 a + 106\right)\cdot 149^{3} + \left(29 a + 67\right)\cdot 149^{4} +O(149^{5})\) 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.