Properties

Label 3.2e4_3e2_7e4.42t37.1c2
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{4} \cdot 3^{2} \cdot 7^{4}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$345744= 2^{4} \cdot 3^{2} \cdot 7^{4} $
Artin number field: Splitting field of $f= x^{7} - 7 x^{5} + 21 x^{3} - 14 x^{2} - 7 x + 4 $ 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 $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{3} + 4 x + 17 $
Roots:
$r_{ 1 }$ $=$ $ 4 a^{2} + 13 a + 10 + \left(4 a^{2} + 18 a + 1\right)\cdot 19 + \left(8 a^{2} + 14 a + 17\right)\cdot 19^{2} + \left(14 a^{2} + 17 a + 9\right)\cdot 19^{3} + \left(7 a^{2} + 11 a + 18\right)\cdot 19^{4} + \left(9 a^{2} + 12 a + 8\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 8 + 6\cdot 19 + 16\cdot 19^{2} + 9\cdot 19^{3} + 2\cdot 19^{4} + 6\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 17 a^{2} + 3 a + 18 + \left(9 a^{2} + a + 14\right)\cdot 19 + \left(a^{2} + 16 a + 9\right)\cdot 19^{2} + \left(7 a^{2} + 14 a + 12\right)\cdot 19^{3} + \left(14 a^{2} + 4 a + 1\right)\cdot 19^{4} + \left(17 a^{2} + 16 a + 17\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 9 a^{2} + 15 a + 17 + \left(4 a^{2} + 13 a + 14\right)\cdot 19 + \left(5 a^{2} + 15 a + 2\right)\cdot 19^{2} + \left(8 a^{2} + 8 a + 6\right)\cdot 19^{3} + \left(4 a^{2} + 7 a + 3\right)\cdot 19^{4} + \left(13 a^{2} + 6 a\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 6 a^{2} + 10 a + 9 + \left(10 a^{2} + 5 a + 11\right)\cdot 19 + \left(5 a^{2} + 7 a + 3\right)\cdot 19^{2} + \left(15 a^{2} + 11 a + 12\right)\cdot 19^{3} + \left(6 a^{2} + 18 a + 9\right)\cdot 19^{4} + \left(15 a^{2} + 18 a + 18\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 16 a^{2} + 14 a + 9 + \left(12 a^{2} + 17 a + 16\right)\cdot 19 + \left(13 a^{2} + 13 a + 16\right)\cdot 19^{2} + \left(9 a^{2} + 9 a + 6\right)\cdot 19^{3} + \left(12 a^{2} + 6 a + 9\right)\cdot 19^{4} + \left(4 a + 9\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 5 a^{2} + 2 a + 5 + \left(15 a^{2} + 10\right)\cdot 19 + \left(3 a^{2} + 8 a + 9\right)\cdot 19^{2} + \left(2 a^{2} + 13 a + 18\right)\cdot 19^{3} + \left(11 a^{2} + 7 a + 11\right)\cdot 19^{4} + \left(17 a + 15\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$

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

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

Character values on conjugacy classes

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