Properties

Label 3.803912.42t37.b.b
Dimension $3$
Group $\GL(3,2)$
Conductor $803912$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $3$
Group: $\GL(3,2)$
Conductor: \(803912\)\(\medspace = 2^{3} \cdot 317^{2} \)
Artin stem field: Galois closure of 7.3.6431296.2
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 7.3.6431296.2

Defining polynomial

$f(x)$$=$ \( x^{7} - 3x^{6} + 3x^{5} - x^{4} - 5x^{3} + 5x^{2} + 3x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.