Properties

Label 10.3e14_29e6.30t88.1c1
Dimension 10
Group $A_6$
Conductor $ 3^{14} \cdot 29^{6}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$10$
Group:$A_6$
Conductor:$2845021504820049= 3^{14} \cdot 29^{6} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{3} - 3 x + 4 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_6$
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 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 17\cdot 19 + 19^{2} + 6\cdot 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 a + 14 + \left(a + 18\right)\cdot 19 + \left(8 a + 6\right)\cdot 19^{2} + \left(16 a + 12\right)\cdot 19^{3} + \left(5 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 a + 1 + \left(9 a + 14\right)\cdot 19 + \left(5 a + 9\right)\cdot 19^{2} + \left(5 a + 8\right)\cdot 19^{3} + \left(9 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 + 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 16 a + 4 + \left(9 a + 1\right)\cdot 19 + \left(13 a + 6\right)\cdot 19^{2} + \left(13 a + 8\right)\cdot 19^{3} + \left(9 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 4 a + 10 + \left(17 a + 5\right)\cdot 19 + \left(10 a + 13\right)\cdot 19^{2} + \left(2 a + 1\right)\cdot 19^{3} + \left(13 a + 11\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$10$
$45$$2$$(1,2)(3,4)$$-2$
$40$$3$$(1,2,3)(4,5,6)$$1$
$40$$3$$(1,2,3)$$1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$72$$5$$(1,2,3,4,5)$$0$
$72$$5$$(1,3,4,5,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.