Properties

Label 3.7542728.42t37.a.a
Dimension $3$
Group $\GL(3,2)$
Conductor $7542728$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $3$
Group: $\GL(3,2)$
Conductor: \(7542728\)\(\medspace = 2^{3} \cdot 971^{2} \)
Artin stem field: Galois closure of 7.3.60341824.1
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 7.3.60341824.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.