Properties

Label 6.6037e2.7t5.1c1
Dimension 6
Group $\GL(3,2)$
Conductor $ 6037^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$\GL(3,2)$
Conductor:$36445369= 6037^{2} $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 4 x^{5} - x^{4} + 4 x^{3} - 4 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\GL(3,2)$
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 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{3} + 2 x + 11 $
Roots:
$r_{ 1 }$ $=$ $ 2 a^{2} + 9 a + 1 + \left(7 a + 8\right)\cdot 13 + \left(a + 10\right)\cdot 13^{2} + \left(6 a^{2} + 11 a + 8\right)\cdot 13^{3} + \left(12 a^{2} + 4 a + 6\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 + 9\cdot 13 + 5\cdot 13^{2} + 6\cdot 13^{3} + 12\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 a^{2} + 9 a + 6 + \left(2 a^{2} + 3 a + 9\right)\cdot 13 + \left(a^{2} + 5 a + 10\right)\cdot 13^{2} + \left(4 a^{2} + 11 a + 6\right)\cdot 13^{3} + \left(5 a^{2} + 10 a + 8\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 10 a^{2} + 5 a + 7 + \left(a^{2} + 6 a + 8\right)\cdot 13 + \left(5 a^{2} + 11\right)\cdot 13^{2} + \left(3 a^{2} + 5\right)\cdot 13^{3} + \left(11 a^{2} + 10 a + 3\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 3 a^{2} + 9 a + 11 + \left(10 a^{2} + 11 a + 12\right)\cdot 13 + \left(5 a^{2} + 10 a\right)\cdot 13^{2} + \left(10 a^{2} + 11 a + 6\right)\cdot 13^{3} + \left(10 a^{2} + 7 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 8 a^{2} + 8 a + 9 + \left(2 a^{2} + 6 a + 2\right)\cdot 13 + \left(7 a^{2} + 7\right)\cdot 13^{2} + \left(9 a^{2} + 3 a\right)\cdot 13^{3} + \left(2 a^{2} + 11\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 10 a^{2} + 12 a + 7 + \left(8 a^{2} + 2 a\right)\cdot 13 + \left(6 a^{2} + 7 a + 5\right)\cdot 13^{2} + \left(5 a^{2} + a + 4\right)\cdot 13^{3} + \left(9 a^{2} + 5 a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

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

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

Character values on conjugacy classes

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