# Properties

 Label 21.109...601.84.a.a Dimension $21$ Group $S_7$ Conductor $1.097\times 10^{53}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $21$ Group: $S_7$ Conductor: $$109\!\cdots\!601$$$$\medspace = 7^{16} \cdot 8951^{10}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 7.1.438599.1 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 7.1.438599.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $$x^{2} + 78x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$74 a + 6 + \left(24 a + 68\right)\cdot 79 + \left(20 a + 74\right)\cdot 79^{2} + \left(16 a + 31\right)\cdot 79^{3} + \left(38 a + 59\right)\cdot 79^{4} +O(79^{5})$$ 74*a + 6 + (24*a + 68)*79 + (20*a + 74)*79^2 + (16*a + 31)*79^3 + (38*a + 59)*79^4+O(79^5) $r_{ 2 }$ $=$ $$27 + 12\cdot 79 + 23\cdot 79^{2} + 18\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})$$ 27 + 12*79 + 23*79^2 + 18*79^3 + 10*79^4+O(79^5) $r_{ 3 }$ $=$ $$5 a + 1 + \left(54 a + 19\right)\cdot 79 + \left(58 a + 70\right)\cdot 79^{2} + \left(62 a + 27\right)\cdot 79^{3} + \left(40 a + 2\right)\cdot 79^{4} +O(79^{5})$$ 5*a + 1 + (54*a + 19)*79 + (58*a + 70)*79^2 + (62*a + 27)*79^3 + (40*a + 2)*79^4+O(79^5) $r_{ 4 }$ $=$ $$76 a + 44 + \left(68 a + 31\right)\cdot 79 + \left(41 a + 19\right)\cdot 79^{2} + \left(56 a + 43\right)\cdot 79^{3} + \left(a + 62\right)\cdot 79^{4} +O(79^{5})$$ 76*a + 44 + (68*a + 31)*79 + (41*a + 19)*79^2 + (56*a + 43)*79^3 + (a + 62)*79^4+O(79^5) $r_{ 5 }$ $=$ $$74 + 41\cdot 79 + 30\cdot 79^{2} + 2\cdot 79^{3} + 73\cdot 79^{4} +O(79^{5})$$ 74 + 41*79 + 30*79^2 + 2*79^3 + 73*79^4+O(79^5) $r_{ 6 }$ $=$ $$46 + 39\cdot 79 + 26\cdot 79^{2} + 55\cdot 79^{3} + 21\cdot 79^{4} +O(79^{5})$$ 46 + 39*79 + 26*79^2 + 55*79^3 + 21*79^4+O(79^5) $r_{ 7 }$ $=$ $$3 a + 41 + \left(10 a + 24\right)\cdot 79 + \left(37 a + 71\right)\cdot 79^{2} + \left(22 a + 57\right)\cdot 79^{3} + \left(77 a + 7\right)\cdot 79^{4} +O(79^{5})$$ 3*a + 41 + (10*a + 24)*79 + (37*a + 71)*79^2 + (22*a + 57)*79^3 + (77*a + 7)*79^4+O(79^5)

## 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.