Properties

Label 3.2e3_457e2.42t37.1c2
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{3} \cdot 457^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$1670792= 2^{3} \cdot 457^{2} $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 2 x^{5} + 2 x^{4} - 4 x^{3} + 4 x^{2} - 4 $ 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 $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{3} + x + 28 $
Roots:
$r_{ 1 }$ $=$ $ a^{2} + 9 a + \left(25 a^{2} + 19 a + 23\right)\cdot 31 + \left(12 a^{2} + 25 a + 16\right)\cdot 31^{2} + \left(25 a^{2} + 20 a + 19\right)\cdot 31^{3} + \left(24 a^{2} + 6 a + 19\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 a^{2} + 2 a + 7 + \left(28 a^{2} + 9 a + 14\right)\cdot 31 + \left(25 a^{2} + 9 a + 17\right)\cdot 31^{2} + \left(21 a^{2} + 6 a + 28\right)\cdot 31^{3} + \left(21 a^{2} + 14 a + 22\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 13 a^{2} + 23 a + 29 + \left(14 a^{2} + 20 a + 4\right)\cdot 31 + \left(a + 21\right)\cdot 31^{2} + \left(21 a^{2} + 26 a + 17\right)\cdot 31^{3} + \left(30 a^{2} + 14 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a^{2} + 6 a + 25 + \left(19 a^{2} + a + 28\right)\cdot 31 + \left(4 a^{2} + 20 a + 23\right)\cdot 31^{2} + \left(19 a^{2} + 29 a + 26\right)\cdot 31^{3} + \left(9 a^{2} + a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 17 a^{2} + 28 a + 21 + \left(a + 6\right)\cdot 31 + \left(29 a^{2} + 7 a + 17\right)\cdot 31^{2} + \left(27 a^{2} + 2 a\right)\cdot 31^{3} + \left(6 a^{2} + 21 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 5 + 26\cdot 31 + 5\cdot 31^{2} + 12\cdot 31^{3} + 27\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 13 a^{2} + 25 a + 8 + \left(5 a^{2} + 9 a + 20\right)\cdot 31 + \left(20 a^{2} + 29 a + 21\right)\cdot 31^{2} + \left(8 a^{2} + 7 a + 18\right)\cdot 31^{3} + \left(30 a^{2} + 3 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

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