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

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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

SizeOrderAction 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.