Properties

Label 35.149...991.70.a.a
Dimension $35$
Group $S_7$
Conductor $1.496\times 10^{78}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(149\!\cdots\!991\)\(\medspace = 11^{24} \cdot 3511^{15}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.424831.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: odd
Determinant: 1.3511.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.424831.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} - x^{4} + 2x^{3} - x^{2} + 2x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 163 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 163 }$: \( x^{2} + 159x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 141 a + 19 + \left(98 a + 97\right)\cdot 163 + \left(38 a + 61\right)\cdot 163^{2} + 52\cdot 163^{3} + \left(67 a + 62\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 35 a + 152 + \left(91 a + 114\right)\cdot 163 + \left(59 a + 130\right)\cdot 163^{2} + \left(52 a + 130\right)\cdot 163^{3} + \left(153 a + 93\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 22 a + 94 + \left(64 a + 25\right)\cdot 163 + \left(124 a + 117\right)\cdot 163^{2} + \left(162 a + 14\right)\cdot 163^{3} + \left(95 a + 4\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 128 a + 129 + \left(71 a + 118\right)\cdot 163 + \left(103 a + 114\right)\cdot 163^{2} + \left(110 a + 117\right)\cdot 163^{3} + \left(9 a + 2\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 92 a + 70 + \left(122 a + 57\right)\cdot 163 + \left(55 a + 5\right)\cdot 163^{2} + \left(146 a + 113\right)\cdot 163^{3} + \left(83 a + 37\right)\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 77 + 108\cdot 163 + 116\cdot 163^{2} + 69\cdot 163^{3} + 61\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 71 a + 112 + \left(40 a + 129\right)\cdot 163 + \left(107 a + 105\right)\cdot 163^{2} + \left(16 a + 153\right)\cdot 163^{3} + \left(79 a + 63\right)\cdot 163^{4} +O(163^{5})\) 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$$1$$()$$35$
$21$$2$$(1,2)$$5$
$105$$2$$(1,2)(3,4)(5,6)$$1$
$105$$2$$(1,2)(3,4)$$-1$
$70$$3$$(1,2,3)$$-1$
$280$$3$$(1,2,3)(4,5,6)$$-1$
$210$$4$$(1,2,3,4)$$-1$
$630$$4$$(1,2,3,4)(5,6)$$1$
$504$$5$$(1,2,3,4,5)$$0$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$420$$6$$(1,2,3)(4,5)$$-1$
$840$$6$$(1,2,3,4,5,6)$$1$
$720$$7$$(1,2,3,4,5,6,7)$$0$
$504$$10$$(1,2,3,4,5)(6,7)$$0$
$420$$12$$(1,2,3,4)(5,6,7)$$-1$

The blue line marks the conjugacy class containing complex conjugation.