# Properties

 Label 10.14117306610774528.70.a.b Dimension 10 Group $A_7$ Conductor $2^{9} \cdot 3^{14} \cdot 7^{8}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $10$ Group: $A_7$ Conductor: $14117306610774528= 2^{9} \cdot 3^{14} \cdot 7^{8}$ Artin number field: Splitting field of 7.3.112021056.1 defined by $f= x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 70 Parity: Even Determinant: 1.1.1t1.a.a Projective image: $A_7$ Projective field: Galois closure of 7.3.112021056.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 659 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 659 }$: $x^{2} + 655 x + 2$
Roots:
 $r_{ 1 }$ $=$ $84 a + 206 + \left(212 a + 614\right)\cdot 659 + \left(642 a + 490\right)\cdot 659^{2} + \left(642 a + 529\right)\cdot 659^{3} + \left(484 a + 480\right)\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 2 }$ $=$ $385 + 268\cdot 659 + 129\cdot 659^{2} + 397\cdot 659^{3} + 485\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 3 }$ $=$ $158 + 292\cdot 659 + 202\cdot 659^{2} + 538\cdot 659^{3} + 347\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 4 }$ $=$ $575 a + 542 + \left(446 a + 60\right)\cdot 659 + \left(16 a + 212\right)\cdot 659^{2} + \left(16 a + 482\right)\cdot 659^{3} + \left(174 a + 459\right)\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 5 }$ $=$ $141 a + 604 + \left(154 a + 480\right)\cdot 659 + \left(465 a + 24\right)\cdot 659^{2} + \left(510 a + 317\right)\cdot 659^{3} + \left(575 a + 322\right)\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 6 }$ $=$ $518 a + 509 + \left(504 a + 297\right)\cdot 659 + \left(193 a + 413\right)\cdot 659^{2} + \left(148 a + 576\right)\cdot 659^{3} + \left(83 a + 137\right)\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 7 }$ $=$ $235 + 621\cdot 659 + 503\cdot 659^{2} + 453\cdot 659^{3} + 401\cdot 659^{4} +O\left(659^{ 5 }\right)$

### 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 $1$ $1$ $()$ $10$ $105$ $2$ $(1,2)(3,4)$ $-2$ $70$ $3$ $(1,2,3)$ $1$ $280$ $3$ $(1,2,3)(4,5,6)$ $1$ $630$ $4$ $(1,2,3,4)(5,6)$ $0$ $504$ $5$ $(1,2,3,4,5)$ $0$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $1$ $360$ $7$ $(1,2,3,4,5,6,7)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $360$ $7$ $(1,3,4,5,6,7,2)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
The blue line marks the conjugacy class containing complex conjugation.