# Properties

 Label 14.936...681.21t38.a.a Dimension $14$ Group $S_7$ Conductor $9.363\times 10^{21}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $14$ Group: $S_7$ Conductor: $$936\!\cdots\!681$$$$\medspace = 277^{4} \cdot 1123^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 7.1.311071.1 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 7.1.311071.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $$x^{2} + 96x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$8 + 75\cdot 97 + 45\cdot 97^{2} + 83\cdot 97^{3} + 21\cdot 97^{4} +O(97^{5})$$ 8 + 75*97 + 45*97^2 + 83*97^3 + 21*97^4+O(97^5) $r_{ 2 }$ $=$ $$54 a + 47 + \left(16 a + 94\right)\cdot 97 + \left(9 a + 77\right)\cdot 97^{2} + \left(54 a + 62\right)\cdot 97^{3} + \left(20 a + 93\right)\cdot 97^{4} +O(97^{5})$$ 54*a + 47 + (16*a + 94)*97 + (9*a + 77)*97^2 + (54*a + 62)*97^3 + (20*a + 93)*97^4+O(97^5) $r_{ 3 }$ $=$ $$43 a + 4 + \left(80 a + 57\right)\cdot 97 + \left(87 a + 70\right)\cdot 97^{2} + \left(42 a + 10\right)\cdot 97^{3} + \left(76 a + 60\right)\cdot 97^{4} +O(97^{5})$$ 43*a + 4 + (80*a + 57)*97 + (87*a + 70)*97^2 + (42*a + 10)*97^3 + (76*a + 60)*97^4+O(97^5) $r_{ 4 }$ $=$ $$87 a + 25 + \left(64 a + 27\right)\cdot 97 + \left(4 a + 58\right)\cdot 97^{2} + \left(29 a + 58\right)\cdot 97^{3} + \left(53 a + 63\right)\cdot 97^{4} +O(97^{5})$$ 87*a + 25 + (64*a + 27)*97 + (4*a + 58)*97^2 + (29*a + 58)*97^3 + (53*a + 63)*97^4+O(97^5) $r_{ 5 }$ $=$ $$10 a + 15 + \left(32 a + 5\right)\cdot 97 + \left(92 a + 95\right)\cdot 97^{2} + \left(67 a + 82\right)\cdot 97^{3} + \left(43 a + 87\right)\cdot 97^{4} +O(97^{5})$$ 10*a + 15 + (32*a + 5)*97 + (92*a + 95)*97^2 + (67*a + 82)*97^3 + (43*a + 87)*97^4+O(97^5) $r_{ 6 }$ $=$ $$41 + 97 + 15\cdot 97^{2} + 97^{3} + 41\cdot 97^{4} +O(97^{5})$$ 41 + 97 + 15*97^2 + 97^3 + 41*97^4+O(97^5) $r_{ 7 }$ $=$ $$55 + 30\cdot 97 + 25\cdot 97^{2} + 88\cdot 97^{3} + 19\cdot 97^{4} +O(97^{5})$$ 55 + 30*97 + 25*97^2 + 88*97^3 + 19*97^4+O(97^5)

## 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.