Properties

Label 7.676...336.8t37.a.a
Dimension $7$
Group $\GL(3,2)$
Conductor $6.764\times 10^{14}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $7$
Group: $\GL(3,2)$
Conductor: \(676433957622336\)\(\medspace = 2^{6} \cdot 3^{6} \cdot 347^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.624200256.2
Galois orbit size: $1$
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 7.3.624200256.2

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.