# Properties

 Label 3.2e3_73e2.7t3.1c2 Dimension 3 Group $C_7:C_3$ Conductor $2^{3} \cdot 73^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $3$ Group: $C_7:C_3$ Conductor: $42632= 2^{3} \cdot 73^{2}$ Artin number field: Splitting field of $f= x^{7} - 8 x^{5} - 2 x^{4} + 16 x^{3} + 6 x^{2} - 6 x - 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_7:C_3$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{3} + 2 x + 9$
Roots:
 $r_{ 1 }$ $=$ $9 a + 2 + \left(a^{2} + 2\right)\cdot 11 + \left(a^{2} + 5 a\right)\cdot 11^{2} + \left(2 a^{2} + 3 a + 4\right)\cdot 11^{3} + \left(8 a^{2} + a + 6\right)\cdot 11^{4} + \left(8 a^{2} + 7 a\right)\cdot 11^{5} + 7\cdot 11^{6} + \left(a^{2} + 9 a + 9\right)\cdot 11^{7} + \left(5 a^{2} + 6 a\right)\cdot 11^{8} + \left(5 a^{2} + 5 a + 7\right)\cdot 11^{9} + \left(4 a^{2} + 6 a\right)\cdot 11^{10} + \left(a^{2} + 3 a + 8\right)\cdot 11^{11} + \left(4 a^{2} + 5 a\right)\cdot 11^{12} + \left(6 a^{2} + 3 a + 6\right)\cdot 11^{13} + \left(7 a^{2} + 10 a + 6\right)\cdot 11^{14} +O\left(11^{ 15 }\right)$ $r_{ 2 }$ $=$ $2 a^{2} + 4 a + 1 + \left(6 a^{2} + 3 a + 9\right)\cdot 11 + \left(6 a^{2} + 3\right)\cdot 11^{2} + \left(6 a^{2} + 3 a + 6\right)\cdot 11^{3} + \left(4 a^{2} + 4 a + 1\right)\cdot 11^{4} + \left(6 a^{2} + 8 a + 1\right)\cdot 11^{5} + \left(8 a^{2} + 9 a + 10\right)\cdot 11^{6} + \left(9 a^{2} + 6 a + 2\right)\cdot 11^{7} + \left(7 a^{2} + 7 a + 8\right)\cdot 11^{8} + \left(5 a^{2} + 5 a + 3\right)\cdot 11^{9} + \left(7 a + 6\right)\cdot 11^{10} + \left(5 a^{2} + 8 a + 5\right)\cdot 11^{11} + \left(7 a^{2} + 7 a + 1\right)\cdot 11^{12} + 2\cdot 11^{13} + \left(10 a^{2} + 8 a + 6\right)\cdot 11^{14} +O\left(11^{ 15 }\right)$ $r_{ 3 }$ $=$ $2 + 10\cdot 11 + 5\cdot 11^{3} + 10\cdot 11^{4} + 9\cdot 11^{5} + 3\cdot 11^{6} + 2\cdot 11^{7} + 7\cdot 11^{8} + 11^{9} + 8\cdot 11^{10} + 4\cdot 11^{11} + 10\cdot 11^{12} + 10\cdot 11^{13} + 11^{14} +O\left(11^{ 15 }\right)$ $r_{ 4 }$ $=$ $4 a^{2} + 4 a + 10 + \left(2 a^{2} + 10 a + 9\right)\cdot 11 + \left(2 a^{2} + 4 a + 3\right)\cdot 11^{2} + \left(3 a^{2} + 8 a + 1\right)\cdot 11^{3} + \left(5 a^{2} + 10 a + 8\right)\cdot 11^{4} + \left(7 a^{2} + 7 a + 6\right)\cdot 11^{5} + \left(7 a^{2} + 3 a + 6\right)\cdot 11^{6} + \left(3 a^{2} + 4 a + 10\right)\cdot 11^{7} + \left(8 a^{2} + 5 a + 10\right)\cdot 11^{8} + \left(10 a^{2} + 10 a + 2\right)\cdot 11^{9} + \left(7 a^{2} + 4 a + 2\right)\cdot 11^{10} + \left(10 a^{2} + a + 10\right)\cdot 11^{11} + \left(2 a^{2} + 7 a + 8\right)\cdot 11^{12} + \left(4 a + 2\right)\cdot 11^{13} + \left(8 a^{2} + 3 a + 6\right)\cdot 11^{14} +O\left(11^{ 15 }\right)$ $r_{ 5 }$ $=$ $4 a + 1 + \left(6 a^{2} + 2 a\right)\cdot 11 + \left(5 a^{2} + 10 a + 1\right)\cdot 11^{2} + \left(4 a^{2} + 7 a + 3\right)\cdot 11^{3} + \left(2 a^{2} + 9 a + 4\right)\cdot 11^{4} + \left(a + 4\right)\cdot 11^{5} + \left(3 a^{2} + 7 a\right)\cdot 11^{6} + \left(9 a^{2} + 5 a + 7\right)\cdot 11^{7} + \left(5 a^{2} + 4 a + 7\right)\cdot 11^{8} + \left(6 a^{2} + 10 a + 4\right)\cdot 11^{9} + \left(3 a^{2} + 8 a + 7\right)\cdot 11^{10} + \left(a^{2} + a + 8\right)\cdot 11^{11} + \left(10 a^{2} + 7 a + 3\right)\cdot 11^{12} + \left(10 a^{2} + 2 a + 2\right)\cdot 11^{13} + \left(8 a + 4\right)\cdot 11^{14} +O\left(11^{ 15 }\right)$ $r_{ 6 }$ $=$ $9 a^{2} + 9 a + 3 + \left(3 a^{2} + 6 a + 2\right)\cdot 11 + \left(3 a^{2} + 5 a + 3\right)\cdot 11^{2} + \left(2 a^{2} + 4 a + 4\right)\cdot 11^{3} + \left(9 a^{2} + 5 a\right)\cdot 11^{4} + \left(6 a^{2} + 6 a + 9\right)\cdot 11^{5} + a^{2}11^{6} + \left(6 a + 1\right)\cdot 11^{7} + \left(9 a^{2} + 7 a + 6\right)\cdot 11^{8} + \left(10 a^{2} + 10 a + 10\right)\cdot 11^{9} + \left(5 a^{2} + 7 a + 9\right)\cdot 11^{10} + \left(4 a^{2} + 9 a + 4\right)\cdot 11^{11} + \left(10 a^{2} + 8 a + 5\right)\cdot 11^{12} + \left(3 a^{2} + 6 a + 6\right)\cdot 11^{13} + \left(4 a^{2} + 3 a + 9\right)\cdot 11^{14} +O\left(11^{ 15 }\right)$ $r_{ 7 }$ $=$ $7 a^{2} + 3 a + 3 + \left(2 a^{2} + 9 a + 10\right)\cdot 11 + \left(3 a^{2} + 6 a + 8\right)\cdot 11^{2} + \left(3 a^{2} + 5 a + 8\right)\cdot 11^{3} + \left(3 a^{2} + a + 1\right)\cdot 11^{4} + \left(3 a^{2} + a + 1\right)\cdot 11^{5} + 4\cdot 11^{6} + \left(9 a^{2} + a + 10\right)\cdot 11^{7} + \left(7 a^{2} + a + 2\right)\cdot 11^{8} + \left(4 a^{2} + a + 2\right)\cdot 11^{9} + \left(10 a^{2} + 8 a + 9\right)\cdot 11^{10} + \left(9 a^{2} + 7 a + 1\right)\cdot 11^{11} + \left(8 a^{2} + 7 a + 2\right)\cdot 11^{12} + \left(10 a^{2} + 3 a + 2\right)\cdot 11^{13} + \left(a^{2} + 10 a + 9\right)\cdot 11^{14} +O\left(11^{ 15 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $3$ $7$ $3$ $(1,2,5)(3,6,4)$ $0$ $7$ $3$ $(1,5,2)(3,4,6)$ $0$ $3$ $7$ $(1,5,7,3,6,2,4)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $3$ $7$ $(1,3,4,7,2,5,6)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
The blue line marks the conjugacy class containing complex conjugation.