Properties

Label 10.237...336.30t88.a.a
Dimension $10$
Group $A_6$
Conductor $2.371\times 10^{16}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $10$
Group: $A_6$
Conductor: \(23713122135310336\)\(\medspace = 2^{18} \cdot 67^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.287296.1
Galois orbit size: $1$
Smallest permutation container: $A_6$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_6$
Projective stem field: Galois closure of 6.2.287296.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{4} + x^{2} - 2x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 28 a + 5 + \left(26 a + 29\right)\cdot 31 + \left(6 a + 1\right)\cdot 31^{2} + \left(20 a + 13\right)\cdot 31^{3} + \left(12 a + 4\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 28\cdot 31 + 22\cdot 31^{2} + 15\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 26 + \left(18 a + 4\right)\cdot 31 + \left(a + 22\right)\cdot 31^{2} + \left(6 a + 21\right)\cdot 31^{3} + \left(4 a + 3\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 30 + \left(4 a + 23\right)\cdot 31 + \left(24 a + 19\right)\cdot 31^{2} + \left(10 a + 15\right)\cdot 31^{3} + \left(18 a + 9\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 a + 13 + \left(12 a + 1\right)\cdot 31 + \left(29 a + 7\right)\cdot 31^{2} + \left(24 a + 1\right)\cdot 31^{3} + \left(26 a + 6\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 + 5\cdot 31 + 19\cdot 31^{2} + 25\cdot 31^{3} + 30\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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.