# Properties

 Label 14.388625298846706628890624.21t33.a.a Dimension 14 Group $A_7$ Conductor $2^{12} \cdot 7^{6} \cdot 73^{8}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $14$ Group: $A_7$ Conductor: $388625298846706628890624= 2^{12} \cdot 7^{6} \cdot 73^{8}$ Artin number field: Splitting field of 7.3.16711744.1 defined by $f= x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_7$ Parity: Even Determinant: 1.1.1t1.a.a Projective image: $A_7$ Projective field: Galois closure of 7.3.16711744.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$: $x^{2} + 220 x + 2$
Roots:
 $r_{ 1 }$ $=$ $128 + 168\cdot 227 + 137\cdot 227^{2} + 129\cdot 227^{3} + 178\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 2 }$ $=$ $55 + 149\cdot 227 + 170\cdot 227^{2} + 56\cdot 227^{3} + 186\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 3 }$ $=$ $15 + 157\cdot 227 + 55\cdot 227^{2} + 95\cdot 227^{3} + 2\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 4 }$ $=$ $77 a + 157 + \left(121 a + 116\right)\cdot 227 + \left(167 a + 170\right)\cdot 227^{2} + \left(3 a + 69\right)\cdot 227^{3} + \left(33 a + 29\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 5 }$ $=$ $150 a + 15 + \left(105 a + 208\right)\cdot 227 + \left(59 a + 86\right)\cdot 227^{2} + \left(223 a + 155\right)\cdot 227^{3} + \left(193 a + 29\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 6 }$ $=$ $190 a + 59 + \left(137 a + 7\right)\cdot 227 + \left(131 a + 92\right)\cdot 227^{2} + \left(194 a + 39\right)\cdot 227^{3} + \left(133 a + 210\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 7 }$ $=$ $37 a + 27 + \left(89 a + 101\right)\cdot 227 + \left(95 a + 194\right)\cdot 227^{2} + \left(32 a + 134\right)\cdot 227^{3} + \left(93 a + 44\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

 Cycle notation $(3,4,5,6,7)$ $(1,2,3)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $14$ $105$ $2$ $(1,2)(3,4)$ $2$ $70$ $3$ $(1,2,3)$ $2$ $280$ $3$ $(1,2,3)(4,5,6)$ $-1$ $630$ $4$ $(1,2,3,4)(5,6)$ $0$ $504$ $5$ $(1,2,3,4,5)$ $-1$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $2$ $360$ $7$ $(1,2,3,4,5,6,7)$ $0$ $360$ $7$ $(1,3,4,5,6,7,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.