Properties

Label 2.2951.9t3.a.c
Dimension $2$
Group $D_{9}$
Conductor $2951$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{9}$
Conductor: \(2951\)\(\medspace = 13 \cdot 227 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.75836247976801.1
Galois orbit size: $3$
Smallest permutation container: $D_{9}$
Parity: odd
Determinant: 1.2951.2t1.a.a
Projective image: $D_9$
Projective stem field: Galois closure of 9.1.75836247976801.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 4x^{8} + 9x^{7} - 37x^{6} + 187x^{5} - 499x^{4} + 843x^{3} - 734x^{2} + 546x - 13 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{3} + x + 35 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 10 a^{2} + 37 a + 9 + \left(11 a^{2} + 5 a + 38\right)\cdot 41 + \left(16 a + 15\right)\cdot 41^{2} + \left(10 a^{2} + 19 a + 21\right)\cdot 41^{3} + \left(16 a^{2} + 26 a + 3\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 20 a^{2} + 38 a + 33 + \left(12 a^{2} + 30 a + 15\right)\cdot 41 + \left(34 a^{2} + 12 a + 39\right)\cdot 41^{2} + \left(18 a^{2} + 26 a + 22\right)\cdot 41^{3} + \left(30 a^{2} + 28 a + 11\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 15 a^{2} + 4 a + 3 + \left(10 a^{2} + 10 a + 37\right)\cdot 41 + \left(26 a^{2} + 32 a + 39\right)\cdot 41^{2} + \left(40 a^{2} + 30 a + 1\right)\cdot 41^{3} + \left(20 a^{2} + 16 a + 30\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 25 a^{2} + 29 a + 19 + \left(13 a^{2} + 25 a + 12\right)\cdot 41 + \left(14 a^{2} + 29 a + 25\right)\cdot 41^{2} + \left(9 a^{2} + 37 a + 34\right)\cdot 41^{3} + \left(4 a^{2} + 7 a + 22\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 4 a^{2} + 19 a + 36 + \left(14 a^{2} + 20 a + 16\right)\cdot 41 + \left(40 a^{2} + 23 a + 2\right)\cdot 41^{2} + \left(8 a^{2} + 13 a + 30\right)\cdot 41^{3} + \left(2 a^{2} + 6 a + 33\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 34 a^{2} + 39 a + 2 + \left(35 a^{2} + 3 a + 13\right)\cdot 41 + \left(35 a^{2} + 15 a + 5\right)\cdot 41^{2} + \left(36 a^{2} + 10 a + 13\right)\cdot 41^{3} + \left(34 a^{2} + 23 a + 39\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 33 a^{2} + 39 a + 15 + \left(35 a^{2} + 26 a + 40\right)\cdot 41 + \left(19 a^{2} + 34 a + 21\right)\cdot 41^{2} + \left(4 a^{2} + 40 a + 32\right)\cdot 41^{3} + \left(26 a^{2} + 19\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 17 a^{2} + 25 a + 31 + \left(14 a^{2} + 30 a + 30\right)\cdot 41 + \left(7 a^{2} + 4 a + 7\right)\cdot 41^{2} + \left(13 a^{2} + a + 19\right)\cdot 41^{3} + \left(8 a^{2} + 6 a + 10\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 6 a^{2} + 16 a + 20 + \left(16 a^{2} + 9 a\right)\cdot 41 + \left(26 a^{2} + 36 a + 6\right)\cdot 41^{2} + \left(21 a^{2} + 24 a + 29\right)\cdot 41^{3} + \left(20 a^{2} + 6 a + 33\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$2$
$9$$2$$(1,2)(3,6)(4,5)(8,9)$$0$
$2$$3$$(1,4,9)(2,8,5)(3,7,6)$$-1$
$2$$9$$(1,5,7,4,2,6,9,8,3)$$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$
$2$$9$$(1,7,2,9,3,5,4,6,8)$$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$
$2$$9$$(1,2,3,4,8,7,9,5,6)$$\zeta_{9}^{5} + \zeta_{9}^{4}$

The blue line marks the conjugacy class containing complex conjugation.