Properties

Label 1.37.9t1.a
Dimension $1$
Group $C_9$
Conductor $37$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_9$
Conductor:\(37\)
Artin number field: Galois closure of 9.9.3512479453921.1
Galois orbit size: $6$
Smallest permutation container: $C_9$
Parity: even
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Galois action

Roots of defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(1,6,8,2,3,7,9,4,5)$
$(1,9,2)(3,6,4)(5,7,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character values
$c1$ $c2$ $c3$ $c4$ $c5$ $c6$
$1$ $1$ $()$ $1$ $1$ $1$ $1$ $1$ $1$
$1$ $3$ $(1,2,9)(3,4,6)(5,8,7)$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$
$1$ $3$ $(1,9,2)(3,6,4)(5,7,8)$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$
$1$ $9$ $(1,6,8,2,3,7,9,4,5)$ $\zeta_{9}$ $\zeta_{9}^{2}$ $\zeta_{9}^{4}$ $\zeta_{9}^{5}$ $-\zeta_{9}^{4} - \zeta_{9}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$
$1$ $9$ $(1,8,3,9,5,6,2,7,4)$ $\zeta_{9}^{2}$ $\zeta_{9}^{4}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $\zeta_{9}$ $\zeta_{9}^{5}$ $-\zeta_{9}^{4} - \zeta_{9}$
$1$ $9$ $(1,3,5,2,4,8,9,6,7)$ $\zeta_{9}^{4}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $-\zeta_{9}^{4} - \zeta_{9}$ $\zeta_{9}^{2}$ $\zeta_{9}$ $\zeta_{9}^{5}$
$1$ $9$ $(1,7,6,9,8,4,2,5,3)$ $\zeta_{9}^{5}$ $\zeta_{9}$ $\zeta_{9}^{2}$ $-\zeta_{9}^{4} - \zeta_{9}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $\zeta_{9}^{4}$
$1$ $9$ $(1,4,7,2,6,5,9,3,8)$ $-\zeta_{9}^{4} - \zeta_{9}$ $\zeta_{9}^{5}$ $\zeta_{9}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $\zeta_{9}^{4}$ $\zeta_{9}^{2}$
$1$ $9$ $(1,5,4,9,7,3,2,8,6)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $-\zeta_{9}^{4} - \zeta_{9}$ $\zeta_{9}^{5}$ $\zeta_{9}^{4}$ $\zeta_{9}^{2}$ $\zeta_{9}$
The blue line marks the conjugacy class containing complex conjugation.