Properties

Label 3.299475.42t37.a.b
Dimension $3$
Group $\GL(3,2)$
Conductor $299475$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $3$
Group: $\GL(3,2)$
Conductor: \(299475\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 11^{3} \)
Artin stem field: Galois closure of 7.3.9965030625.2
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 7.3.9965030625.2

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} - 9x^{5} + 21x^{4} - 3x^{3} - 23x^{2} + 10x + 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.