Properties

Label 35.867...736.70.a.a
Dimension $35$
Group $S_7$
Conductor $8.675\times 10^{65}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(867\!\cdots\!736\)\(\medspace = 2^{97} \cdot 3^{77}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.1451188224.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: odd
Determinant: 1.24.2t1.b.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.1451188224.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 27x^{3} - 54x^{2} + 42x - 12 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 271 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 271 }$: \( x^{2} + 269x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 125 a + 78 + \left(60 a + 145\right)\cdot 271 + \left(51 a + 155\right)\cdot 271^{2} + \left(22 a + 25\right)\cdot 271^{3} + \left(65 a + 192\right)\cdot 271^{4} +O(271^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 146 a + 57 + \left(210 a + 141\right)\cdot 271 + \left(219 a + 197\right)\cdot 271^{2} + \left(248 a + 18\right)\cdot 271^{3} + \left(205 a + 29\right)\cdot 271^{4} +O(271^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 162 a + 174 + \left(207 a + 74\right)\cdot 271 + \left(194 a + 137\right)\cdot 271^{2} + \left(265 a + 70\right)\cdot 271^{3} + \left(134 a + 216\right)\cdot 271^{4} +O(271^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 94 + 9\cdot 271 + 63\cdot 271^{2} + 11\cdot 271^{3} + 215\cdot 271^{4} +O(271^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 109 a + 227 + \left(63 a + 56\right)\cdot 271 + \left(76 a + 48\right)\cdot 271^{2} + \left(5 a + 136\right)\cdot 271^{3} + \left(136 a + 220\right)\cdot 271^{4} +O(271^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 131 + 72\cdot 271 + 119\cdot 271^{2} + 232\cdot 271^{3} + 26\cdot 271^{4} +O(271^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 54 + 42\cdot 271 + 92\cdot 271^{2} + 47\cdot 271^{3} + 184\cdot 271^{4} +O(271^{5})\) 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.