Properties

Label 35.206...384.70.a.a
Dimension $35$
Group $S_7$
Conductor $2.068\times 10^{59}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(206\!\cdots\!384\)\(\medspace = 2^{75} \cdot 3^{77}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.90699264.1
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 7.1.90699264.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 643 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 643 }$: \( x^{2} + 641x + 11 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 531 a + 389 + \left(248 a + 398\right)\cdot 643 + \left(558 a + 593\right)\cdot 643^{2} + \left(630 a + 619\right)\cdot 643^{3} + \left(436 a + 272\right)\cdot 643^{4} +O(643^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 136 a + 132 + \left(330 a + 449\right)\cdot 643 + \left(384 a + 403\right)\cdot 643^{2} + \left(87 a + 478\right)\cdot 643^{3} + \left(302 a + 302\right)\cdot 643^{4} +O(643^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 507 a + 404 + \left(312 a + 330\right)\cdot 643 + \left(258 a + 199\right)\cdot 643^{2} + \left(555 a + 269\right)\cdot 643^{3} + \left(340 a + 176\right)\cdot 643^{4} +O(643^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 + 129\cdot 643 + 131\cdot 643^{2} + 554\cdot 643^{3} + 454\cdot 643^{4} +O(643^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 9 a + 408 + \left(620 a + 155\right)\cdot 643 + \left(602 a + 241\right)\cdot 643^{2} + \left(398 a + 530\right)\cdot 643^{3} + \left(17 a + 284\right)\cdot 643^{4} +O(643^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 112 a + 165 + \left(394 a + 365\right)\cdot 643 + \left(84 a + 175\right)\cdot 643^{2} + \left(12 a + 37\right)\cdot 643^{3} + \left(206 a + 516\right)\cdot 643^{4} +O(643^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 634 a + 426 + \left(22 a + 100\right)\cdot 643 + \left(40 a + 184\right)\cdot 643^{2} + \left(244 a + 82\right)\cdot 643^{3} + \left(625 a + 564\right)\cdot 643^{4} +O(643^{5})\) 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.