Properties

Label 35.995...351.70.a.a
Dimension $35$
Group $S_7$
Conductor $9.956\times 10^{83}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(995\!\cdots\!351\)\(\medspace = 11^{15} \cdot 97^{15} \cdot 373^{15} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.397991.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: odd
Determinant: 1.397991.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.397991.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{5} + 3x^{3} - x^{2} - x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: \( x^{2} + 82x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 + 32\cdot 83 + 6\cdot 83^{2} + 41\cdot 83^{3} + 26\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 37 + 31\cdot 83 + 6\cdot 83^{2} + 37\cdot 83^{3} + 10\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 79 a + 76 + 32\cdot 83 + \left(52 a + 63\right)\cdot 83^{2} + \left(27 a + 23\right)\cdot 83^{3} + \left(63 a + 16\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 72 + \left(82 a + 37\right)\cdot 83 + \left(30 a + 31\right)\cdot 83^{2} + \left(55 a + 82\right)\cdot 83^{3} + \left(19 a + 51\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 8 a + 76 + \left(51 a + 65\right)\cdot 83 + \left(45 a + 24\right)\cdot 83^{2} + \left(52 a + 71\right)\cdot 83^{3} + \left(18 a + 4\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 68 + 22\cdot 83 + 14\cdot 83^{2} + 81\cdot 83^{3} + 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 75 a + 1 + \left(31 a + 26\right)\cdot 83 + \left(37 a + 19\right)\cdot 83^{2} + \left(30 a + 78\right)\cdot 83^{3} + \left(64 a + 53\right)\cdot 83^{4} +O(83^{5})\) 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.