Properties

Label 35.629...343.70.a.a
Dimension $35$
Group $S_7$
Conductor $6.292\times 10^{75}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(629\!\cdots\!343\)\(\medspace = 7^{18} \cdot 31^{15} \cdot 353^{15}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.536207.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: odd
Determinant: 1.10943.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.536207.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: \( x^{2} + 190x + 19 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 60 a + 1 + \left(105 a + 139\right)\cdot 191 + \left(176 a + 128\right)\cdot 191^{2} + \left(42 a + 3\right)\cdot 191^{3} + \left(20 a + 178\right)\cdot 191^{4} +O(191^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 108 a + 37 + \left(23 a + 160\right)\cdot 191 + \left(40 a + 82\right)\cdot 191^{2} + \left(179 a + 169\right)\cdot 191^{3} + \left(76 a + 182\right)\cdot 191^{4} +O(191^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 45 + 72\cdot 191 + 169\cdot 191^{2} + 35\cdot 191^{3} + 33\cdot 191^{4} +O(191^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 83 a + 145 + \left(167 a + 75\right)\cdot 191 + \left(150 a + 99\right)\cdot 191^{2} + \left(11 a + 117\right)\cdot 191^{3} + \left(114 a + 80\right)\cdot 191^{4} +O(191^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 + 50\cdot 191 + 99\cdot 191^{2} + 181\cdot 191^{3} + 41\cdot 191^{4} +O(191^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 131 a + 61 + \left(85 a + 184\right)\cdot 191 + \left(14 a + 8\right)\cdot 191^{2} + \left(148 a + 61\right)\cdot 191^{3} + \left(170 a + 155\right)\cdot 191^{4} +O(191^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 84 + 82\cdot 191 + 175\cdot 191^{2} + 3\cdot 191^{3} + 92\cdot 191^{4} +O(191^{5})\) 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.