Properties

Label 3.2e4_937e2.42t37.1c2
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{4} \cdot 937^{2}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$14047504= 2^{4} \cdot 937^{2} $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 3 x^{5} - 5 x^{4} + 5 x^{3} + 7 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 $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{3} + 2 x + 11 $
Roots:
$r_{ 1 }$ $=$ $ 10 a^{2} + 6 a + 12 + \left(6 a^{2} + 4 a + 9\right)\cdot 13 + \left(7 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(9 a^{2} + a\right)\cdot 13^{3} + \left(7 a^{2} + 4 a + 9\right)\cdot 13^{4} + \left(6 a^{2} + 12\right)\cdot 13^{5} + \left(7 a^{2} + 11 a + 10\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 12 a^{2} + 6 a + 8 + \left(5 a + 9\right)\cdot 13 + \left(3 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(6 a^{2} + 4\right)\cdot 13^{3} + \left(7 a^{2} + a + 6\right)\cdot 13^{4} + \left(6 a^{2} + 11 a + 10\right)\cdot 13^{5} + \left(4 a^{2} + 2 a + 12\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 3 + 11\cdot 13 + 10\cdot 13^{3} + 13^{4} + 9\cdot 13^{5} + 2\cdot 13^{6} +O\left(13^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 12 a^{2} + 4 a + 8 + \left(3 a^{2} + 2 a\right)\cdot 13 + \left(7 a^{2} + 6\right)\cdot 13^{2} + \left(8 a^{2} + 8 a + 7\right)\cdot 13^{3} + \left(5 a^{2} + 10 a + 12\right)\cdot 13^{4} + \left(7 a^{2} + 2 a + 2\right)\cdot 13^{5} + \left(11 a^{2} + 3 a + 9\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$
$r_{ 5 }$ $=$ $ a^{2} + a + \left(4 a^{2} + 12 a + 2\right)\cdot 13 + \left(8 a^{2} + 6 a + 10\right)\cdot 13^{2} + \left(8 a^{2} + 10 a + 7\right)\cdot 13^{3} + \left(2 a^{2} + 9 a + 6\right)\cdot 13^{4} + \left(3 a^{2} + 9 a + 12\right)\cdot 13^{5} + \left(2 a^{2} + 6 a + 3\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 2 a^{2} + 3 a + 12 + \left(8 a^{2} + 5 a + 1\right)\cdot 13 + \left(2 a^{2} + 4 a + 4\right)\cdot 13^{2} + \left(11 a^{2} + 4 a + 2\right)\cdot 13^{3} + \left(12 a^{2} + a + 9\right)\cdot 13^{4} + \left(11 a^{2} + 12 a + 4\right)\cdot 13^{5} + \left(9 a^{2} + 6 a + 11\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 2 a^{2} + 6 a + 10 + \left(2 a^{2} + 9 a + 3\right)\cdot 13 + \left(10 a^{2} + 10 a + 8\right)\cdot 13^{2} + \left(7 a^{2} + 6\right)\cdot 13^{3} + \left(2 a^{2} + 12 a + 6\right)\cdot 13^{4} + \left(3 a^{2} + 2 a + 12\right)\cdot 13^{5} + \left(3 a^{2} + 8 a\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$

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

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

Character values on conjugacy classes

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