Properties

Label 21.255...176.84.b.a
Dimension $21$
Group $S_7$
Conductor $2.555\times 10^{39}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $21$
Group: $S_7$
Conductor: \(255\!\cdots\!176\)\(\medspace = 2^{58} \cdot 3^{46}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.241864704.2
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 7.1.241864704.2

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: \( x^{2} + 149x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 102 + 81\cdot 151 + 14\cdot 151^{2} + 114\cdot 151^{3} + 106\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 66 a + 48 + \left(89 a + 135\right)\cdot 151 + \left(10 a + 61\right)\cdot 151^{2} + \left(123 a + 12\right)\cdot 151^{3} + \left(61 a + 86\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 72 + 54\cdot 151 + 103\cdot 151^{2} + 147\cdot 151^{3} + 37\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 26 + 52\cdot 151 + 8\cdot 151^{2} + 43\cdot 151^{3} + 46\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 85 a + 29 + \left(61 a + 97\right)\cdot 151 + \left(140 a + 144\right)\cdot 151^{2} + \left(27 a + 96\right)\cdot 151^{3} + \left(89 a + 86\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 83 + 58\cdot 151 + 116\cdot 151^{2} + 130\cdot 151^{3} + 14\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 96 + 124\cdot 151 + 3\cdot 151^{2} + 59\cdot 151^{3} + 74\cdot 151^{4} +O(151^{5})\) 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.