Properties

Label 6.1814078464.7t5.a.a
Dimension $6$
Group $\GL(3,2)$
Conductor $1814078464$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $\GL(3,2)$
Conductor: \(1814078464\)\(\medspace = 2^{10} \cdot 11^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.1814078464.1
Galois orbit size: $1$
Smallest permutation container: $\GL(3,2)$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\GL(3,2)$
Projective stem field: Galois closure of 7.3.1814078464.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} + 2x^{5} - 12x^{4} - 14x^{3} + 10x^{2} + 10x - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{3} + 4x + 17 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.