Properties

Label 35.642...536.126.a.a
Dimension $35$
Group $S_7$
Conductor $6.426\times 10^{64}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(642\!\cdots\!536\)\(\medspace = 2^{98} \cdot 3^{74}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.241864704.1
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 7.1.241864704.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 3x^{5} - 4x^{4} + 5x^{3} - 6x^{2} + 7x - 8 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 557 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 557 }$: \( x^{2} + 553x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 62 a + 548 + \left(460 a + 127\right)\cdot 557 + \left(309 a + 40\right)\cdot 557^{2} + \left(318 a + 472\right)\cdot 557^{3} + \left(484 a + 347\right)\cdot 557^{4} +O(557^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 542 a + 168 + \left(436 a + 109\right)\cdot 557 + \left(503 a + 179\right)\cdot 557^{2} + \left(548 a + 180\right)\cdot 557^{3} + \left(523 a + 350\right)\cdot 557^{4} +O(557^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 420 + 47\cdot 557 + 197\cdot 557^{2} + 416\cdot 557^{3} + 260\cdot 557^{4} +O(557^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 495 a + 239 + \left(96 a + 235\right)\cdot 557 + \left(247 a + 262\right)\cdot 557^{2} + \left(238 a + 322\right)\cdot 557^{3} + \left(72 a + 296\right)\cdot 557^{4} +O(557^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 183 + 406\cdot 557 + 392\cdot 557^{2} + 5\cdot 557^{3} + 451\cdot 557^{4} +O(557^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 + 543\cdot 557 + 512\cdot 557^{2} + 72\cdot 557^{3} + 295\cdot 557^{4} +O(557^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 15 a + 108 + \left(120 a + 201\right)\cdot 557 + \left(53 a + 86\right)\cdot 557^{2} + \left(8 a + 201\right)\cdot 557^{3} + \left(33 a + 226\right)\cdot 557^{4} +O(557^{5})\) 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.