Properties

Label 14.254...881.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $2.547\times 10^{21}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(254\!\cdots\!881\)\(\medspace = 277^{4} \cdot 811^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.224647.1
Galois orbit size: $1$
Smallest permutation container: 21T38
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.224647.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{5} - x^{4} + 2x^{3} + 2x^{2} - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: \( x^{2} + 149x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 142 a + 8 + \left(17 a + 84\right)\cdot 151 + \left(49 a + 63\right)\cdot 151^{2} + \left(61 a + 45\right)\cdot 151^{3} + \left(140 a + 126\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 138\cdot 151 + 38\cdot 151^{2} + 62\cdot 151^{3} + 91\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 95 a + 133 + \left(7 a + 97\right)\cdot 151 + \left(14 a + 53\right)\cdot 151^{2} + \left(93 a + 149\right)\cdot 151^{3} + \left(44 a + 107\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 128 a + 93 + \left(143 a + 139\right)\cdot 151 + \left(44 a + 141\right)\cdot 151^{2} + \left(91 a + 110\right)\cdot 151^{3} + \left(35 a + 150\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 56 a + 21 + \left(143 a + 18\right)\cdot 151 + \left(136 a + 74\right)\cdot 151^{2} + \left(57 a + 19\right)\cdot 151^{3} + \left(106 a + 104\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 9 a + 141 + \left(133 a + 128\right)\cdot 151 + \left(101 a + 143\right)\cdot 151^{2} + \left(89 a + 118\right)\cdot 151^{3} + \left(10 a + 43\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 23 a + 47 + \left(7 a + 148\right)\cdot 151 + \left(106 a + 87\right)\cdot 151^{2} + \left(59 a + 97\right)\cdot 151^{3} + \left(115 a + 130\right)\cdot 151^{4} +O(151^{5})\) 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$$1$$()$$14$
$21$$2$$(1,2)$$6$
$105$$2$$(1,2)(3,4)(5,6)$$2$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$2$
$280$$3$$(1,2,3)(4,5,6)$$-1$
$210$$4$$(1,2,3,4)$$0$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$-1$
$210$$6$$(1,2,3)(4,5)(6,7)$$2$
$420$$6$$(1,2,3)(4,5)$$0$
$840$$6$$(1,2,3,4,5,6)$$-1$
$720$$7$$(1,2,3,4,5,6,7)$$0$
$504$$10$$(1,2,3,4,5)(6,7)$$1$
$420$$12$$(1,2,3,4)(5,6,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.