Properties

Label 3.16128256.42t37.a.b
Dimension $3$
Group $\GL(3,2)$
Conductor $16128256$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $3$
Group: $\GL(3,2)$
Conductor: \(16128256\)\(\medspace = 2^{8} \cdot 251^{2} \)
Artin stem field: Galois closure of 7.3.64513024.2
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 7.3.64513024.2

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{3} + 2x + 11 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 11 a^{2} + a + 11 + \left(a + 10\right)\cdot 13 + \left(5 a^{2} + 3 a + 3\right)\cdot 13^{2} + \left(2 a^{2} + 3\right)\cdot 13^{3} + \left(9 a^{2} + a + 12\right)\cdot 13^{4} + \left(5 a^{2} + 9 a\right)\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a^{2} + 3 a + 10 + \left(11 a^{2} + 12 a + 7\right)\cdot 13 + \left(4 a^{2} + 8 a + 3\right)\cdot 13^{2} + \left(11 a^{2} + 10 a + 2\right)\cdot 13^{3} + \left(6 a^{2} + 6 a + 9\right)\cdot 13^{4} + \left(7 a^{2} + 2 a + 7\right)\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 7 a^{2} + 4 a + \left(5 a^{2} + 6 a + 4\right)\cdot 13 + \left(10 a^{2} + 6 a\right)\cdot 13^{2} + \left(4 a^{2} + 4 a + 3\right)\cdot 13^{3} + \left(10 a^{2} + 2 a + 9\right)\cdot 13^{4} + \left(8 a^{2} + 10 a + 12\right)\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 + 7\cdot 13 + 10\cdot 13^{2} + 9\cdot 13^{3} + 4\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a^{2} + 8 a + 12 + \left(10 a^{2} + 3 a + 5\right)\cdot 13 + \left(8 a^{2} + 6 a + 2\right)\cdot 13^{2} + \left(9 a^{2} + 10 a + 5\right)\cdot 13^{3} + \left(4 a^{2} + 2 a + 10\right)\cdot 13^{4} + \left(6 a^{2} + 4 a\right)\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a^{2} + a + 12 + \left(10 a^{2} + 3 a + 5\right)\cdot 13 + \left(6 a^{2} + 8\right)\cdot 13^{2} + \left(11 a^{2} + 11 a + 7\right)\cdot 13^{3} + \left(10 a^{2} + 7 a + 5\right)\cdot 13^{4} + \left(10 a^{2} + 11 a + 2\right)\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 8 a^{2} + 9 a + 7 + \left(12 a + 10\right)\cdot 13 + \left(3 a^{2} + 9\right)\cdot 13^{2} + \left(12 a^{2} + 2 a + 7\right)\cdot 13^{3} + \left(9 a^{2} + 5 a + 4\right)\cdot 13^{4} + \left(12 a^{2} + a + 10\right)\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.