Properties

Label 3.2e3_661e2.42t37.1c1
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{3} \cdot 661^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$3495368= 2^{3} \cdot 661^{2} $
Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{5} + 3 x^{4} - 5 x^{3} + 7 x^{2} - 5 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 $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
$r_{ 1 }$ $=$ $ 11 a^{2} + 24 a + 18 + \left(19 a^{2} + 21 a + 2\right)\cdot 29 + \left(22 a^{2} + 24 a + 4\right)\cdot 29^{2} + \left(11 a^{2} + 28 a + 26\right)\cdot 29^{3} + \left(24 a^{2} + 15 a + 11\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 + 3\cdot 29 + 10\cdot 29^{2} + 2\cdot 29^{3} + 12\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 25 a^{2} + 16 a + 5 + \left(13 a^{2} + 21 a + 21\right)\cdot 29 + \left(14 a^{2} + 19 a + 22\right)\cdot 29^{2} + \left(3 a^{2} + a + 12\right)\cdot 29^{3} + \left(a^{2} + 9 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 19 a^{2} + 12 a + 26 + \left(17 a^{2} + 8 a + 6\right)\cdot 29 + \left(2 a^{2} + 25 a + 26\right)\cdot 29^{2} + \left(16 a^{2} + 16 a + 19\right)\cdot 29^{3} + \left(13 a^{2} + 19 a + 5\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 21 a^{2} + 9 a + 12 + \left(5 a^{2} + 19 a + 13\right)\cdot 29 + \left(8 a^{2} + 5 a + 23\right)\cdot 29^{2} + \left(22 a^{2} + 20\right)\cdot 29^{3} + \left(7 a^{2} + 11 a + 18\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 26 a^{2} + 25 a + 9 + \left(3 a^{2} + 16 a + 1\right)\cdot 29 + \left(27 a^{2} + 27 a + 10\right)\cdot 29^{2} + \left(23 a^{2} + 28 a + 13\right)\cdot 29^{3} + \left(25 a^{2} + a + 23\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 14 a^{2} + a + \left(26 a^{2} + 28 a + 9\right)\cdot 29 + \left(11 a^{2} + 12 a + 19\right)\cdot 29^{2} + \left(9 a^{2} + 10 a + 20\right)\cdot 29^{3} + \left(14 a^{2} + 6\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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