Properties

Label 3.11e3_31e2.42t37.1c1
Dimension 3
Group $\GL(3,2)$
Conductor $ 11^{3} \cdot 31^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$1279091= 11^{3} \cdot 31^{2} $
Artin number field: Splitting field of $f= x^{7} - 11 x^{5} - 22 x^{4} + 33 x^{2} + 33 x + 11 $ 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 $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
$r_{ 1 }$ $=$ $ a^{2} + 3 a + 23 + \left(15 a^{2} + 9 a + 15\right)\cdot 29 + \left(6 a^{2} + 8 a + 7\right)\cdot 29^{2} + \left(25 a^{2} + 27 a + 8\right)\cdot 29^{3} + \left(5 a^{2} + 22 a + 20\right)\cdot 29^{4} + \left(a^{2} + 13 a + 10\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 9 a^{2} + 28 a + 8 + \left(25 a^{2} + 8 a + 15\right)\cdot 29 + \left(4 a^{2} + 23 a + 21\right)\cdot 29^{2} + \left(18 a^{2} + 28 a + 15\right)\cdot 29^{3} + \left(26 a^{2} + 19 a\right)\cdot 29^{4} + \left(15 a^{2} + 2 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 10 a^{2} + 28 a + 19 + \left(21 a^{2} + 22 a + 19\right)\cdot 29 + \left(2 a^{2} + 2 a + 18\right)\cdot 29^{2} + \left(a^{2} + 25 a + 2\right)\cdot 29^{3} + \left(5 a^{2} + 18 a + 20\right)\cdot 29^{4} + \left(2 a^{2} + 9 a + 28\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 26 a + 12 + \left(17 a^{2} + 25 a + 18\right)\cdot 29 + \left(21 a^{2} + 4 a + 27\right)\cdot 29^{2} + \left(17 a^{2} + 5 a + 7\right)\cdot 29^{3} + \left(12 a^{2} + 17 a\right)\cdot 29^{4} + \left(13 a^{2} + 9 a + 27\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 5 + 10\cdot 29 + 16\cdot 29^{2} + 14\cdot 29^{3} + 9\cdot 29^{4} + 11\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 10 a^{2} + 2 a + 19 + \left(11 a^{2} + 26 a + 25\right)\cdot 29 + \left(21 a^{2} + 2 a + 4\right)\cdot 29^{2} + \left(9 a^{2} + 4 a + 14\right)\cdot 29^{3} + \left(26 a^{2} + 19 a + 19\right)\cdot 29^{4} + \left(10 a^{2} + 16 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 28 a^{2} + 1 + \left(25 a^{2} + 23 a + 11\right)\cdot 29 + \left(15 a + 19\right)\cdot 29^{2} + \left(15 a^{2} + 25 a + 23\right)\cdot 29^{3} + \left(10 a^{2} + 17 a + 16\right)\cdot 29^{4} + \left(14 a^{2} + 5 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$

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

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

Character values on conjugacy classes

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