# Properties

 Label 14.152...625.21t38.a.a Dimension $14$ Group $S_7$ Conductor $1.529\times 10^{22}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $14$ Group: $S_7$ Conductor: $$152\!\cdots\!625$$$$\medspace = 5^{4} \cdot 53^{4} \cdot 1327^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 7.1.351655.1 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 7.1.351655.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 659 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$198 + 23\cdot 659 + 152\cdot 659^{2} + 170\cdot 659^{3} + 92\cdot 659^{4} +O(659^{5})$$ 198 + 23*659 + 152*659^2 + 170*659^3 + 92*659^4+O(659^5) $r_{ 2 }$ $=$ $$307 + 109\cdot 659 + 175\cdot 659^{2} + 260\cdot 659^{3} + 652\cdot 659^{4} +O(659^{5})$$ 307 + 109*659 + 175*659^2 + 260*659^3 + 652*659^4+O(659^5) $r_{ 3 }$ $=$ $$312 + 7\cdot 659 + 100\cdot 659^{2} + 302\cdot 659^{3} + 90\cdot 659^{4} +O(659^{5})$$ 312 + 7*659 + 100*659^2 + 302*659^3 + 90*659^4+O(659^5) $r_{ 4 }$ $=$ $$357 + 397\cdot 659 + 594\cdot 659^{2} + 483\cdot 659^{3} + 535\cdot 659^{4} +O(659^{5})$$ 357 + 397*659 + 594*659^2 + 483*659^3 + 535*659^4+O(659^5) $r_{ 5 }$ $=$ $$457 + 9\cdot 659 + 2\cdot 659^{2} + 251\cdot 659^{3} + 433\cdot 659^{4} +O(659^{5})$$ 457 + 9*659 + 2*659^2 + 251*659^3 + 433*659^4+O(659^5) $r_{ 6 }$ $=$ $$480 + 621\cdot 659 + 17\cdot 659^{2} + 489\cdot 659^{3} + 569\cdot 659^{4} +O(659^{5})$$ 480 + 621*659 + 17*659^2 + 489*659^3 + 569*659^4+O(659^5) $r_{ 7 }$ $=$ $$526 + 148\cdot 659 + 276\cdot 659^{2} + 20\cdot 659^{3} + 262\cdot 659^{4} +O(659^{5})$$ 526 + 148*659 + 276*659^2 + 20*659^3 + 262*659^4+O(659^5)

## 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.