Properties

Label 3.2e4_5e2_11e3.42t37.1c2
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{4} \cdot 5^{2} \cdot 11^{3}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$532400= 2^{4} \cdot 5^{2} \cdot 11^{3} $
Artin number field: Splitting field of $f= x^{7} - x^{6} - 9 x^{5} - x^{4} + 8 x^{3} - 12 x^{2} - 12 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 $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{3} + 2 x + 18 $
Roots:
$r_{ 1 }$ $=$ $ 6 a^{2} + a + 1 + \left(9 a^{2} + 20 a + 9\right)\cdot 23 + \left(20 a^{2} + a + 9\right)\cdot 23^{2} + \left(2 a^{2} + 9 a + 4\right)\cdot 23^{3} + \left(2 a^{2} + 10 a + 21\right)\cdot 23^{4} + \left(13 a^{2} + 7 a + 22\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 15 a^{2} + 6 a + 2 + \left(13 a^{2} + 19 a + 14\right)\cdot 23 + \left(12 a^{2} + 4 a + 19\right)\cdot 23^{2} + \left(4 a^{2} + 14 a + 1\right)\cdot 23^{3} + \left(21 a^{2} + 4 a + 5\right)\cdot 23^{4} + \left(10 a^{2} + 3 a + 18\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 9 a^{2} + 21 a + 17 + \left(22 a^{2} + 15 a + 2\right)\cdot 23 + \left(4 a^{2} + 7 a + 17\right)\cdot 23^{2} + \left(10 a^{2} + 20 a + 1\right)\cdot 23^{3} + \left(15 a^{2} + 22 a + 5\right)\cdot 23^{4} + \left(5 a^{2} + 18 a + 11\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 2 a^{2} + 10 a + 11 + \left(2 a^{2} + 7 a + 22\right)\cdot 23 + \left(10 a^{2} + 16 a + 10\right)\cdot 23^{2} + \left(8 a^{2} + 15 a + 19\right)\cdot 23^{3} + \left(14 a^{2} + 17 a + 6\right)\cdot 23^{4} + \left(21 a^{2} + 11\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 7 + 22\cdot 23 + 21\cdot 23^{2} + 10\cdot 23^{3} + 14\cdot 23^{4} + 18\cdot 23^{5} +O\left(23^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 22 a^{2} + 19 a + 19 + \left(9 a^{2} + 10 a + 16\right)\cdot 23 + \left(5 a^{2} + 10 a + 17\right)\cdot 23^{2} + \left(8 a^{2} + 11 a + 6\right)\cdot 23^{3} + \left(9 a^{2} + 18 a + 12\right)\cdot 23^{4} + \left(6 a^{2} + 4\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 15 a^{2} + 12 a + 13 + \left(11 a^{2} + 18 a + 4\right)\cdot 23 + \left(15 a^{2} + 4 a + 18\right)\cdot 23^{2} + \left(11 a^{2} + 21 a\right)\cdot 23^{3} + \left(6 a^{2} + 17 a + 4\right)\cdot 23^{4} + \left(11 a^{2} + 14 a + 5\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$

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

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

Character values on conjugacy classes

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