# Properties

 Label 21.132...249.84.a.a Dimension $21$ Group $S_7$ Conductor $1.330\times 10^{42}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $21$ Group: $S_7$ Conductor: $$132\!\cdots\!249$$$$\medspace = 23^{10} \cdot 709^{10}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 7.5.11561663.1 Galois orbit size: $1$ Smallest permutation container: 84 Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_7$ Projective stem field: Galois closure of 7.5.11561663.1

## Defining polynomial

 $f(x)$ $=$ $$x^{7} - x^{6} - 3x^{5} + 5x^{4} + 2x^{3} - 8x^{2} - 2x + 1$$ x^7 - x^6 - 3*x^5 + 5*x^4 + 2*x^3 - 8*x^2 - 2*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 311 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 311 }$: $$x^{2} + 310x + 17$$

Roots:
 $r_{ 1 }$ $=$ $$247 a + 33 + \left(16 a + 221\right)\cdot 311 + \left(23 a + 182\right)\cdot 311^{2} + \left(118 a + 249\right)\cdot 311^{3} + \left(206 a + 242\right)\cdot 311^{4} +O(311^{5})$$ 247*a + 33 + (16*a + 221)*311 + (23*a + 182)*311^2 + (118*a + 249)*311^3 + (206*a + 242)*311^4+O(311^5) $r_{ 2 }$ $=$ $$27 + 256\cdot 311 + 198\cdot 311^{2} + 92\cdot 311^{3} + 86\cdot 311^{4} +O(311^{5})$$ 27 + 256*311 + 198*311^2 + 92*311^3 + 86*311^4+O(311^5) $r_{ 3 }$ $=$ $$105 + 264\cdot 311 + 16\cdot 311^{2} + 213\cdot 311^{3} + 152\cdot 311^{4} +O(311^{5})$$ 105 + 264*311 + 16*311^2 + 213*311^3 + 152*311^4+O(311^5) $r_{ 4 }$ $=$ $$5 + 126\cdot 311 + 126\cdot 311^{2} + 251\cdot 311^{3} + 32\cdot 311^{4} +O(311^{5})$$ 5 + 126*311 + 126*311^2 + 251*311^3 + 32*311^4+O(311^5) $r_{ 5 }$ $=$ $$259 a + 268 + \left(294 a + 174\right)\cdot 311 + \left(22 a + 245\right)\cdot 311^{2} + \left(90 a + 12\right)\cdot 311^{3} + \left(173 a + 2\right)\cdot 311^{4} +O(311^{5})$$ 259*a + 268 + (294*a + 174)*311 + (22*a + 245)*311^2 + (90*a + 12)*311^3 + (173*a + 2)*311^4+O(311^5) $r_{ 6 }$ $=$ $$64 a + 280 + \left(294 a + 301\right)\cdot 311 + \left(287 a + 188\right)\cdot 311^{2} + \left(192 a + 33\right)\cdot 311^{3} + \left(104 a + 20\right)\cdot 311^{4} +O(311^{5})$$ 64*a + 280 + (294*a + 301)*311 + (287*a + 188)*311^2 + (192*a + 33)*311^3 + (104*a + 20)*311^4+O(311^5) $r_{ 7 }$ $=$ $$52 a + 216 + \left(16 a + 210\right)\cdot 311 + \left(288 a + 284\right)\cdot 311^{2} + \left(220 a + 79\right)\cdot 311^{3} + \left(137 a + 85\right)\cdot 311^{4} +O(311^{5})$$ 52*a + 216 + (16*a + 210)*311 + (288*a + 284)*311^2 + (220*a + 79)*311^3 + (137*a + 85)*311^4+O(311^5)

## 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 $1$ $1$ $()$ $21$ $21$ $2$ $(1,2)$ $1$ $105$ $2$ $(1,2)(3,4)(5,6)$ $-3$ $105$ $2$ $(1,2)(3,4)$ $1$ $70$ $3$ $(1,2,3)$ $-3$ $280$ $3$ $(1,2,3)(4,5,6)$ $0$ $210$ $4$ $(1,2,3,4)$ $-1$ $630$ $4$ $(1,2,3,4)(5,6)$ $-1$ $504$ $5$ $(1,2,3,4,5)$ $1$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $1$ $420$ $6$ $(1,2,3)(4,5)$ $1$ $840$ $6$ $(1,2,3,4,5,6)$ $0$ $720$ $7$ $(1,2,3,4,5,6,7)$ $0$ $504$ $10$ $(1,2,3,4,5)(6,7)$ $1$ $420$ $12$ $(1,2,3,4)(5,6,7)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.