Properties

Label 3.2e4_19e2_47e2.42t37.1c1
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{4} \cdot 19^{2} \cdot 47^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$12759184= 2^{4} \cdot 19^{2} \cdot 47^{2} $
Artin number field: Splitting field of $f= x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 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 $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{3} + 6 x + 35 $
Roots:
$r_{ 1 }$ $=$ $ 3 a^{2} + 27 a + 14 + \left(34 a^{2} + 3 a + 28\right)\cdot 37 + \left(35 a^{2} + 2 a + 17\right)\cdot 37^{2} + \left(19 a^{2} + 12 a + 34\right)\cdot 37^{3} + \left(11 a^{2} + 32 a + 3\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 a^{2} + 21 + \left(25 a^{2} + 15 a + 16\right)\cdot 37 + \left(17 a^{2} + 11 a + 4\right)\cdot 37^{2} + \left(7 a^{2} + 15 a + 22\right)\cdot 37^{3} + \left(3 a^{2} + 32 a + 23\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 33 a^{2} + 31 a + 35 + \left(34 a^{2} + 29 a + 17\right)\cdot 37 + \left(21 a^{2} + 17 a + 21\right)\cdot 37^{2} + \left(33 a^{2} + 23 a + 15\right)\cdot 37^{3} + \left(20 a^{2} + 28 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 26 + 22\cdot 37 + 21\cdot 37^{2} + 11\cdot 37^{3} + 20\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a^{2} + 35 a + 22 + \left(6 a^{2} + 26 a + 27\right)\cdot 37 + \left(20 a^{2} + 4 a + 28\right)\cdot 37^{2} + \left(18 a^{2} + 35 a + 28\right)\cdot 37^{3} + \left(21 a^{2} + 27 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 29 a^{2} + 12 a + 7 + \left(33 a^{2} + 6 a + 27\right)\cdot 37 + \left(17 a^{2} + 30 a + 19\right)\cdot 37^{2} + \left(35 a^{2} + 26 a + 22\right)\cdot 37^{3} + \left(3 a^{2} + 13 a + 10\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 30 a^{2} + 6 a + 23 + \left(13 a^{2} + 29 a + 7\right)\cdot 37 + \left(34 a^{2} + 7 a + 34\right)\cdot 37^{2} + \left(32 a^{2} + 35 a + 12\right)\cdot 37^{3} + \left(12 a^{2} + 12 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

Character values on conjugacy classes

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