Properties

Label 35.193...736.126.a.a
Dimension $35$
Group $S_7$
Conductor $1.935\times 10^{60}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(193\!\cdots\!736\)\(\medspace = 2^{102} \cdot 3^{62}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.161243136.1
Galois orbit size: $1$
Smallest permutation container: 126
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.161243136.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 6x^{5} - 10x^{4} + 14x^{3} - 12x^{2} + 4x + 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 389 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 389 }$: \( x^{2} + 379x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 46 + 318\cdot 389 + 188\cdot 389^{2} + 303\cdot 389^{3} + 325\cdot 389^{4} +O(389^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 226 + 81\cdot 389 + 2\cdot 389^{2} + 134\cdot 389^{3} + 3\cdot 389^{4} +O(389^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 78 a + 124 + \left(217 a + 258\right)\cdot 389 + \left(350 a + 367\right)\cdot 389^{2} + \left(324 a + 32\right)\cdot 389^{3} + \left(237 a + 251\right)\cdot 389^{4} +O(389^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 311 a + 126 + \left(171 a + 18\right)\cdot 389 + \left(38 a + 155\right)\cdot 389^{2} + \left(64 a + 208\right)\cdot 389^{3} + \left(151 a + 359\right)\cdot 389^{4} +O(389^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 339 a + 174 + \left(340 a + 227\right)\cdot 389 + \left(254 a + 363\right)\cdot 389^{2} + \left(49 a + 186\right)\cdot 389^{3} + \left(334 a + 140\right)\cdot 389^{4} +O(389^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 21 + 78\cdot 389 + 241\cdot 389^{2} + 261\cdot 389^{3} + 155\cdot 389^{4} +O(389^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 50 a + 63 + \left(48 a + 185\right)\cdot 389 + \left(134 a + 237\right)\cdot 389^{2} + \left(339 a + 39\right)\cdot 389^{3} + \left(54 a + 320\right)\cdot 389^{4} +O(389^{5})\) 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.