Properties

Label 3.2e3_5e2_127e2.42t37.1c2
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{3} \cdot 5^{2} \cdot 127^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$3225800= 2^{3} \cdot 5^{2} \cdot 127^{2} $
Artin number field: Splitting field of $f= x^{7} - x^{6} - 3 x^{5} + 7 x^{4} - 7 x^{3} - x^{2} + 3 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 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{3} + 2 x + 11 $
Roots:
$r_{ 1 }$ $=$ $ 4 a^{2} + 6 a + 12 + \left(5 a^{2} + 10 a + 12\right)\cdot 13 + \left(12 a^{2} + 6 a + 12\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + \left(8 a^{2} + 6 a + 10\right)\cdot 13^{4} + \left(7 a^{2} + 11 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 9 a^{2} + 4 a + 7 + \left(8 a^{2} + 5\right)\cdot 13 + \left(12 a^{2} + 12 a + 11\right)\cdot 13^{2} + \left(2 a^{2} + 2 a + 10\right)\cdot 13^{3} + \left(8 a^{2} + 2 a + 2\right)\cdot 13^{4} + \left(12 a^{2} + 7 a + 3\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 8 + \left(9 a^{2} + 3 a + 1\right)\cdot 13 + \left(12 a^{2} + 9 a + 7\right)\cdot 13^{2} + \left(a + 12\right)\cdot 13^{3} + \left(11 a^{2} + 2 a + 10\right)\cdot 13^{4} + \left(8 a^{2} + 12 a + 6\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 4 a^{2} + 3 a + 9 + \left(8 a^{2} + 9 a\right)\cdot 13 + \left(4 a + 4\right)\cdot 13^{2} + \left(9 a^{2} + 8 a + 10\right)\cdot 13^{3} + \left(6 a^{2} + 8 a\right)\cdot 13^{4} + \left(4 a^{2} + 6 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 9 + 13^{2} + 4\cdot 13^{3} + 12\cdot 13^{4} +O\left(13^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 11 a^{2} + 12 a + 4 + \left(11 a^{2} + 8 a + 4\right)\cdot 13 + \left(5 a^{2} + a + 4\right)\cdot 13^{2} + \left(4 a^{2} + 4 a + 6\right)\cdot 13^{3} + \left(10 a^{2} + 9 a\right)\cdot 13^{4} + \left(9 a^{2} + a + 9\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 11 a^{2} + 8 a + 4 + \left(8 a^{2} + 6 a\right)\cdot 13 + \left(7 a^{2} + 4 a + 11\right)\cdot 13^{2} + \left(7 a^{2} + 10 a + 1\right)\cdot 13^{3} + \left(7 a^{2} + 9 a + 1\right)\cdot 13^{4} + \left(8 a^{2} + 12 a + 3\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$

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

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

Character values on conjugacy classes

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