Properties

Label 3.13e2_109e2.42t37.1c1
Dimension 3
Group $\GL(3,2)$
Conductor $ 13^{2} \cdot 109^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$2007889= 13^{2} \cdot 109^{2} $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} - x^{5} + 4 x^{4} - 3 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $\PSL(2,7)$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $ x^{3} + 2 x + 9 $
Roots:
$r_{ 1 }$ $=$ $ 2 a + 2 + \left(6 a^{2} + 10 a + 8\right)\cdot 11 + \left(10 a^{2} + 4 a\right)\cdot 11^{2} + \left(2 a + 7\right)\cdot 11^{3} + \left(3 a^{2} + 9 a + 4\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 a^{2} + 9 a + 1 + \left(3 a^{2} + 8\right)\cdot 11 + \left(5 a^{2} + 10 a + 9\right)\cdot 11^{2} + \left(3 a^{2} + 10 a\right)\cdot 11^{3} + \left(7 a^{2} + 3 a\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 9 a^{2} + 7 a + 3 + \left(6 a + 1\right)\cdot 11 + \left(2 a^{2} + 5 a + 4\right)\cdot 11^{2} + \left(10 a^{2} + 2 a + 8\right)\cdot 11^{3} + \left(4 a^{2} + 8 a + 10\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a^{2} + 7 a + 3 + \left(a^{2} + 2 a + 1\right)\cdot 11 + \left(7 a^{2} + 3 a + 1\right)\cdot 11^{2} + \left(2 a^{2} + 6 a + 7\right)\cdot 11^{3} + \left(6 a^{2} + 4 a + 9\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 2 a^{2} + 2 a + 1 + \left(4 a^{2} + 5 a + 2\right)\cdot 11 + \left(9 a^{2} + 10\right)\cdot 11^{2} + \left(10 a^{2} + 6 a + 1\right)\cdot 11^{3} + \left(2 a^{2} + 4 a + 8\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 10 a^{2} + 6 a + \left(5 a^{2} + 7 a\right)\cdot 11 + \left(9 a^{2} + 8 a + 8\right)\cdot 11^{2} + \left(4 a^{2} + 4 a + 2\right)\cdot 11^{3} + \left(8 a^{2} + 2 a + 5\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 3 + 11 + 10\cdot 11^{2} + 4\cdot 11^{3} + 5\cdot 11^{4} +O\left(11^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$3$
$21$$2$$(1,2)(5,7)$$-1$
$56$$3$$(1,3,6)(2,5,4)$$0$
$42$$4$$(1,5,6,4)(3,7)$$1$
$24$$7$$(1,7,3,5,6,4,2)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$24$$7$$(1,5,2,3,4,7,6)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
The blue line marks the conjugacy class containing complex conjugation.