# Properties

 Label 14.165...081.21t38.a.a Dimension $14$ Group $S_7$ Conductor $1.654\times 10^{21}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $14$ Group: $S_7$ Conductor: $$165\!\cdots\!081$$$$\medspace = 17^{4} \cdot 11863^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 7.1.201671.1 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 7.1.201671.1

## Defining polynomial

 $f(x)$ $=$ $$x^{7} - x^{6} + 2x^{5} - 2x^{4} + 2x^{3} - 2x^{2} + 2x - 1$$ x^7 - x^6 + 2*x^5 - 2*x^4 + 2*x^3 - 2*x^2 + 2*x - 1 .

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $$x^{2} + 274x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$168 a + 90 + \left(208 a + 2\right)\cdot 277 + \left(236 a + 96\right)\cdot 277^{2} + \left(152 a + 64\right)\cdot 277^{3} + \left(129 a + 87\right)\cdot 277^{4} +O(277^{5})$$ 168*a + 90 + (208*a + 2)*277 + (236*a + 96)*277^2 + (152*a + 64)*277^3 + (129*a + 87)*277^4+O(277^5) $r_{ 2 }$ $=$ $$177 a + 107 + \left(86 a + 169\right)\cdot 277 + \left(32 a + 270\right)\cdot 277^{2} + \left(165 a + 183\right)\cdot 277^{3} + \left(63 a + 168\right)\cdot 277^{4} +O(277^{5})$$ 177*a + 107 + (86*a + 169)*277 + (32*a + 270)*277^2 + (165*a + 183)*277^3 + (63*a + 168)*277^4+O(277^5) $r_{ 3 }$ $=$ $$167 a + 114 + \left(265 a + 45\right)\cdot 277 + \left(177 a + 249\right)\cdot 277^{2} + \left(125 a + 27\right)\cdot 277^{3} + \left(57 a + 75\right)\cdot 277^{4} +O(277^{5})$$ 167*a + 114 + (265*a + 45)*277 + (177*a + 249)*277^2 + (125*a + 27)*277^3 + (57*a + 75)*277^4+O(277^5) $r_{ 4 }$ $=$ $$100 a + 84 + \left(190 a + 252\right)\cdot 277 + \left(244 a + 3\right)\cdot 277^{2} + \left(111 a + 93\right)\cdot 277^{3} + \left(213 a + 194\right)\cdot 277^{4} +O(277^{5})$$ 100*a + 84 + (190*a + 252)*277 + (244*a + 3)*277^2 + (111*a + 93)*277^3 + (213*a + 194)*277^4+O(277^5) $r_{ 5 }$ $=$ $$59 + 57\cdot 277 + 204\cdot 277^{2} + 225\cdot 277^{3} + 137\cdot 277^{4} +O(277^{5})$$ 59 + 57*277 + 204*277^2 + 225*277^3 + 137*277^4+O(277^5) $r_{ 6 }$ $=$ $$109 a + 40 + \left(68 a + 183\right)\cdot 277 + \left(40 a + 43\right)\cdot 277^{2} + \left(124 a + 9\right)\cdot 277^{3} + \left(147 a + 46\right)\cdot 277^{4} +O(277^{5})$$ 109*a + 40 + (68*a + 183)*277 + (40*a + 43)*277^2 + (124*a + 9)*277^3 + (147*a + 46)*277^4+O(277^5) $r_{ 7 }$ $=$ $$110 a + 61 + \left(11 a + 121\right)\cdot 277 + \left(99 a + 240\right)\cdot 277^{2} + \left(151 a + 226\right)\cdot 277^{3} + \left(219 a + 121\right)\cdot 277^{4} +O(277^{5})$$ 110*a + 61 + (11*a + 121)*277 + (99*a + 240)*277^2 + (151*a + 226)*277^3 + (219*a + 121)*277^4+O(277^5)

## 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

 Size Order Action 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.