Properties

Label 35.245...344.70.a.a
Dimension $35$
Group $S_7$
Conductor $2.451\times 10^{59}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(245\!\cdots\!344\)\(\medspace = 2^{80} \cdot 3^{74}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.60466176.2
Galois orbit size: $1$
Smallest permutation container: 70
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.60466176.2

Defining polynomial

$f(x)$$=$ \( x^{7} - 3x^{5} - 2x^{4} + 3x^{3} + 12x^{2} + 19x + 6 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 313 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 313 }$: \( x^{2} + 310x + 10 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 119 + 78\cdot 313 + 158\cdot 313^{2} + 256\cdot 313^{3} + 278\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 68 a + 262 + \left(148 a + 309\right)\cdot 313 + \left(207 a + 252\right)\cdot 313^{2} + \left(199 a + 152\right)\cdot 313^{3} + \left(240 a + 83\right)\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 219 + 150\cdot 313 + 203\cdot 313^{2} + 244\cdot 313^{3} + 119\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 51 a + 281 + \left(203 a + 81\right)\cdot 313 + \left(66 a + 308\right)\cdot 313^{2} + \left(145 a + 67\right)\cdot 313^{3} + \left(177 a + 84\right)\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 245 a + 153 + \left(164 a + 60\right)\cdot 313 + \left(105 a + 101\right)\cdot 313^{2} + \left(113 a + 231\right)\cdot 313^{3} + \left(72 a + 292\right)\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 97 + 243\cdot 313 + 235\cdot 313^{2} + 174\cdot 313^{3} + 234\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 262 a + 121 + \left(109 a + 14\right)\cdot 313 + \left(246 a + 305\right)\cdot 313^{2} + \left(167 a + 123\right)\cdot 313^{3} + \left(135 a + 158\right)\cdot 313^{4} +O(313^{5})\) 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.