Properties

Label 3.2e4_53e2.42t37.1c2
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{4} \cdot 53^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$44944= 2^{4} \cdot 53^{2} $
Artin number field: Splitting field of $f= x^{7} - 7 x^{5} - 10 x^{4} + 3 x^{3} + 3 x + 2 $ 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 $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{3} + 4 x + 17 $
Roots:
$r_{ 1 }$ $=$ $ 18 a^{2} + 15 a + 1 + \left(9 a^{2} + 2 a + 5\right)\cdot 19 + \left(5 a^{2} + 5\right)\cdot 19^{2} + \left(4 a^{2} + 10 a + 13\right)\cdot 19^{3} + \left(13 a^{2} + 17 a + 13\right)\cdot 19^{4} + \left(14 a^{2} + 5 a + 1\right)\cdot 19^{5} + \left(10 a^{2} + 10 a + 7\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 7 + 3\cdot 19 + 16\cdot 19^{2} + 18\cdot 19^{3} + 10\cdot 19^{4} + 10\cdot 19^{5} + 16\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 5 a^{2} + 8 a + 17 + \left(10 a^{2} + 18 a + 5\right)\cdot 19 + \left(14 a^{2} + a + 10\right)\cdot 19^{2} + \left(11 a^{2} + 18 a + 1\right)\cdot 19^{3} + \left(4 a^{2} + 17 a + 16\right)\cdot 19^{4} + \left(11 a^{2} + 11 a + 17\right)\cdot 19^{5} + \left(15 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 18 a^{2} + 11 a + 4 + \left(18 a^{2} + 14 a + 1\right)\cdot 19 + \left(5 a^{2} + a + 1\right)\cdot 19^{2} + \left(13 a^{2} + 2 a + 2\right)\cdot 19^{3} + \left(15 a^{2} + 9\right)\cdot 19^{4} + \left(8 a^{2} + 18 a\right)\cdot 19^{5} + \left(12 a^{2} + 3 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 12 a^{2} + a + 7 + \left(a^{2} + 14 a + 18\right)\cdot 19 + \left(3 a^{2} + 11 a + 5\right)\cdot 19^{2} + \left(8 a^{2} + 11 a + 7\right)\cdot 19^{3} + \left(3 a^{2} + 10 a + 14\right)\cdot 19^{4} + \left(5 a^{2} + 16 a + 9\right)\cdot 19^{5} + \left(16 a^{2} + 10 a + 2\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 8 a^{2} + 7 a + 9 + \left(17 a^{2} + 9 a + 3\right)\cdot 19 + \left(9 a^{2} + 5 a + 5\right)\cdot 19^{2} + \left(16 a^{2} + 5 a + 4\right)\cdot 19^{3} + \left(18 a^{2} + 8 a + 11\right)\cdot 19^{4} + \left(4 a^{2} + 3 a + 15\right)\cdot 19^{5} + \left(9 a^{2} + 4 a + 2\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 15 a^{2} + 15 a + 12 + \left(17 a^{2} + 16 a\right)\cdot 19 + \left(17 a^{2} + 16 a + 13\right)\cdot 19^{2} + \left(2 a^{2} + 9 a + 9\right)\cdot 19^{3} + \left(a^{2} + 2 a\right)\cdot 19^{4} + \left(12 a^{2} + a + 1\right)\cdot 19^{5} + \left(7 a^{2} + 12 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$

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

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

Character values on conjugacy classes

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