Properties

Label 3.8832784.42t37.b.a
Dimension $3$
Group $\GL(3,2)$
Conductor $8832784$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $3$
Group: $\GL(3,2)$
Conductor: \(8832784\)\(\medspace = 2^{4} \cdot 743^{2} \)
Artin stem field: Galois closure of 7.3.35331136.1
Galois orbit size: $2$
Smallest permutation container: $\PSL(2,7)$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\GL(3,2)$
Projective stem field: Galois closure of 7.3.35331136.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} - 3x^{5} + 4x^{4} + 5x^{3} - 3x - 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{3} + 2x + 27 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 18 a^{2} + 12 a + 16 + \left(23 a^{2} + 13 a + 5\right)\cdot 29 + \left(18 a^{2} + 17 a + 11\right)\cdot 29^{2} + \left(20 a^{2} + 23\right)\cdot 29^{3} + \left(25 a^{2} + 22 a + 3\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 a^{2} + 13 a + 6 + \left(7 a^{2} + 5 a + 23\right)\cdot 29 + \left(12 a^{2} + 27 a + 21\right)\cdot 29^{2} + \left(10 a^{2} + 23 a + 9\right)\cdot 29^{3} + \left(15 a^{2} + 8 a + 9\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 a^{2} + 5 a + 6 + \left(23 a^{2} + 24 a + 27\right)\cdot 29 + \left(6 a^{2} + 16 a + 16\right)\cdot 29^{2} + \left(12 a^{2} + 9 a + 3\right)\cdot 29^{3} + \left(21 a^{2} + 2 a + 19\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a^{2} + 19 a + 6 + \left(3 a^{2} + 4 a + 10\right)\cdot 29 + \left(10 a^{2} + 13 a + 21\right)\cdot 29^{2} + \left(16 a^{2} + 26 a + 18\right)\cdot 29^{3} + \left(23 a^{2} + 10 a + 2\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 15 a^{2} + 4 a + 12 + \left(26 a^{2} + 10 a + 9\right)\cdot 29 + \left(26 a^{2} + 13 a + 12\right)\cdot 29^{2} + \left(26 a^{2} + 4 a + 12\right)\cdot 29^{3} + \left(16 a^{2} + 27 a + 11\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 a^{2} + 5 a + 3 + \left(2 a^{2} + 9\right)\cdot 29 + \left(12 a^{2} + 28 a + 14\right)\cdot 29^{2} + \left(21 a + 26\right)\cdot 29^{3} + \left(13 a^{2} + 15 a + 7\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 11 + 2\cdot 29 + 18\cdot 29^{2} + 21\cdot 29^{3} + 3\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$3$
$21$$2$$(2,3)(4,7)$$-1$
$56$$3$$(1,4,7)(2,5,3)$$0$
$42$$4$$(1,5,7,2)(3,6)$$1$
$24$$7$$(1,5,4,7,3,6,2)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$24$$7$$(1,7,2,4,6,5,3)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$

The blue line marks the conjugacy class containing complex conjugation.