Properties

Label 3.2e8_7e4.42t37.1c2
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{8} \cdot 7^{4}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$614656= 2^{8} \cdot 7^{4} $
Artin number field: Splitting field of $f= x^{7} - 7 x^{5} - 14 x^{4} - 7 x^{3} - 7 x + 2 $ 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 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{3} + 2 x + 11 $
Roots:
$r_{ 1 }$ $=$ $ 8 a^{2} + 6 a + 10 + \left(5 a^{2} + 10 a + 5\right)\cdot 13 + \left(7 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(5 a^{2} + 3 a + 4\right)\cdot 13^{3} + \left(2 a^{2} + a + 3\right)\cdot 13^{4} + \left(2 a^{2} + 5 a + 8\right)\cdot 13^{5} + \left(8 a^{2} + 5 a + 1\right)\cdot 13^{6} + \left(4 a + 9\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 12 a + 8 + \left(7 a^{2} + 7 a + 7\right)\cdot 13 + \left(10 a^{2} + 2 a + 8\right)\cdot 13^{2} + \left(4 a^{2} + 7\right)\cdot 13^{3} + 12 a\cdot 13^{4} + \left(6 a^{2} + 9\right)\cdot 13^{5} + \left(9 a^{2} + 7 a + 7\right)\cdot 13^{6} + 2 a\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 5 + 6\cdot 13 + 10\cdot 13^{2} + 6\cdot 13^{3} + 6\cdot 13^{5} + 3\cdot 13^{6} + 7\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 9 a^{2} + 5 a + 11 + \left(6 a^{2} + a + 12\right)\cdot 13 + \left(a^{2} + 6 a + 3\right)\cdot 13^{2} + \left(9 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(2 a^{2} + 7 a + 3\right)\cdot 13^{4} + \left(3 a^{2} + 10 a + 1\right)\cdot 13^{5} + \left(9 a^{2} + 3 a + 3\right)\cdot 13^{6} + \left(7 a^{2} + 8 a + 8\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 12 a^{2} + 8 a + 2 + \left(10 a^{2} + 12 a + 1\right)\cdot 13 + \left(2 a^{2} + 9 a + 10\right)\cdot 13^{2} + \left(3 a + 9\right)\cdot 13^{3} + \left(12 a^{2} + 5 a + 2\right)\cdot 13^{4} + \left(a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(10 a^{2} + 9 a + 8\right)\cdot 13^{6} + \left(11 a + 7\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 5 a^{2} + 10 + \left(8 a^{2} + 12 a + 10\right)\cdot 13 + \left(8 a^{2} + 9 a + 4\right)\cdot 13^{2} + \left(3 a^{2} + 9 a + 1\right)\cdot 13^{3} + \left(11 a^{2} + 12 a + 6\right)\cdot 13^{4} + \left(7 a^{2} + 11 a + 7\right)\cdot 13^{5} + \left(6 a^{2} + 12 a + 12\right)\cdot 13^{6} + \left(4 a^{2} + 5 a + 3\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 5 a^{2} + 8 a + 6 + \left(7 a + 7\right)\cdot 13 + \left(8 a^{2} + a + 9\right)\cdot 13^{2} + \left(2 a^{2} + 9 a + 4\right)\cdot 13^{3} + \left(10 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(4 a^{2} + 6 a + 11\right)\cdot 13^{5} + \left(8 a^{2} + 1\right)\cdot 13^{6} + \left(11 a^{2} + 6 a + 2\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$

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

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

Character values on conjugacy classes

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