Properties

Label 6.28291761.7t5.b.a
Dimension $6$
Group $\GL(3,2)$
Conductor $28291761$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $\GL(3,2)$
Conductor: \(28291761\)\(\medspace = 3^{6} \cdot 197^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.28291761.2
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 7.3.28291761.2

Defining polynomial

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

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

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

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$$3$$(1,4,3)(2,5,6)$$0$
$42$$4$$(1,7)(2,4,6,3)$$0$
$24$$7$$(1,2,5,4,6,3,7)$$-1$
$24$$7$$(1,4,7,5,3,2,6)$$-1$

The blue line marks the conjugacy class containing complex conjugation.