Properties

Label 3.2e3_1237e2.42t37.1c1
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{3} \cdot 1237^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$12241352= 2^{3} \cdot 1237^{2} $
Artin number field: Splitting field of $f= x^{7} + 2 x^{5} - 12 x^{4} + 10 x^{3} - 14 x^{2} + 10 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 $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{3} + x + 14 $
Roots:
$r_{ 1 }$ $=$ $ 5 a^{2} + 4 a + 1 + \left(9 a^{2} + 15 a\right)\cdot 17 + \left(2 a^{2} + 6 a + 16\right)\cdot 17^{2} + \left(6 a^{2} + 14 a + 4\right)\cdot 17^{3} + \left(11 a^{2} + 12 a + 11\right)\cdot 17^{4} + \left(7 a^{2} + a + 13\right)\cdot 17^{5} + \left(10 a^{2} + 15 a + 5\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 12 a^{2} + 10 a + \left(10 a^{2} + a + 1\right)\cdot 17 + \left(2 a^{2} + 11 a + 16\right)\cdot 17^{2} + \left(8 a^{2} + 7 a + 11\right)\cdot 17^{3} + \left(3 a^{2} + 11\right)\cdot 17^{4} + \left(16 a^{2} + 16 a + 13\right)\cdot 17^{5} + \left(5 a^{2} + 3 a + 2\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 16 + 15\cdot 17 + 12\cdot 17^{2} + 5\cdot 17^{3} + 16\cdot 17^{4} + 9\cdot 17^{5} + 14\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 4 }$ $=$ $ a + 14 + \left(a^{2} + 14 a + 12\right)\cdot 17 + \left(7 a^{2} + 5 a + 8\right)\cdot 17^{2} + \left(4 a^{2} + a + 11\right)\cdot 17^{3} + \left(2 a^{2} + 4 a + 3\right)\cdot 17^{4} + \left(7 a^{2} + 16 a + 4\right)\cdot 17^{5} + \left(10 a + 2\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 13 a^{2} + 7 a + \left(4 a^{2} + 15 a + 4\right)\cdot 17 + \left(15 a^{2} + 14 a + 14\right)\cdot 17^{2} + \left(a^{2} + 4 a + 9\right)\cdot 17^{3} + \left(11 a^{2} + 8 a + 9\right)\cdot 17^{4} + \left(4 a^{2} + 7 a + 2\right)\cdot 17^{5} + \left(7 a^{2} + 13 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 4 a^{2} + 9 a + 11 + \left(11 a^{2} + 4 a + 2\right)\cdot 17 + \left(11 a^{2} + 13 a + 6\right)\cdot 17^{2} + \left(10 a^{2} + 10 a + 4\right)\cdot 17^{3} + \left(3 a^{2} + 4 a + 10\right)\cdot 17^{4} + \left(5 a^{2} + 10 a + 8\right)\cdot 17^{5} + \left(9 a^{2} + 9 a + 2\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 3 a + 9 + \left(14 a^{2} + 14\right)\cdot 17 + \left(11 a^{2} + 16 a + 10\right)\cdot 17^{2} + \left(2 a^{2} + 11 a + 2\right)\cdot 17^{3} + \left(2 a^{2} + 3 a + 5\right)\cdot 17^{4} + \left(10 a^{2} + 16 a + 15\right)\cdot 17^{5} + \left(14 a + 4\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$

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

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

Character values on conjugacy classes

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