Properties

Label 35.427...507.70.a.a
Dimension $35$
Group $S_7$
Conductor $4.272\times 10^{83}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(427\!\cdots\!507\)\(\medspace = 389^{15} \cdot 967^{15} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.376163.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: odd
Determinant: 1.376163.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.376163.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: \( x^{2} + 97x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 39 + \left(69 a + 61\right)\cdot 101 + \left(36 a + 2\right)\cdot 101^{2} + \left(35 a + 48\right)\cdot 101^{3} + \left(13 a + 73\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 87 a + 70 + \left(91 a + 9\right)\cdot 101 + \left(46 a + 40\right)\cdot 101^{2} + \left(76 a + 51\right)\cdot 101^{3} + \left(67 a + 84\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 50 + 33\cdot 101 + 58\cdot 101^{2} + 33\cdot 101^{3} + 70\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 80 a + 22 + \left(31 a + 14\right)\cdot 101 + \left(64 a + 80\right)\cdot 101^{2} + \left(65 a + 51\right)\cdot 101^{3} + \left(87 a + 91\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 42 a + 21 + \left(70 a + 29\right)\cdot 101 + \left(88 a + 2\right)\cdot 101^{2} + \left(92 a + 15\right)\cdot 101^{3} + \left(43 a + 63\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 14 a + 14 + \left(9 a + 88\right)\cdot 101 + \left(54 a + 34\right)\cdot 101^{2} + \left(24 a + 7\right)\cdot 101^{3} + \left(33 a + 77\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 59 a + 88 + \left(30 a + 66\right)\cdot 101 + \left(12 a + 84\right)\cdot 101^{2} + \left(8 a + 95\right)\cdot 101^{3} + \left(57 a + 44\right)\cdot 101^{4} +O(101^{5})\) 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.