Properties

Label 5.23066015625.6t15.d.a
Dimension $5$
Group $A_6$
Conductor $23066015625$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $A_6$
Conductor: \(23066015625\)\(\medspace = 3^{10} \cdot 5^{8} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.2562890625.1
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 6.2.2562890625.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} + 15x^{4} - 25x^{3} - 45x + 60 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{2} + 82x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 75 a + 87 + \left(5 a + 44\right)\cdot 89 + \left(56 a + 10\right)\cdot 89^{2} + \left(53 a + 31\right)\cdot 89^{3} + \left(75 a + 1\right)\cdot 89^{4} + \left(52 a + 13\right)\cdot 89^{5} +O(89^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 45 + \left(22 a + 45\right)\cdot 89 + \left(20 a + 25\right)\cdot 89^{2} + \left(72 a + 17\right)\cdot 89^{3} + \left(69 a + 77\right)\cdot 89^{4} + \left(31 a + 83\right)\cdot 89^{5} +O(89^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 86 a + 66 + \left(66 a + 18\right)\cdot 89 + \left(68 a + 56\right)\cdot 89^{2} + \left(16 a + 57\right)\cdot 89^{3} + \left(19 a + 48\right)\cdot 89^{4} + \left(57 a + 58\right)\cdot 89^{5} +O(89^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 68 + 87\cdot 89 + 21\cdot 89^{2} + 12\cdot 89^{3} + 4\cdot 89^{4} + 38\cdot 89^{5} +O(89^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 14 a + 78 + \left(83 a + 10\right)\cdot 89 + \left(32 a + 41\right)\cdot 89^{2} + \left(35 a + 83\right)\cdot 89^{3} + \left(13 a + 31\right)\cdot 89^{4} + \left(36 a + 40\right)\cdot 89^{5} +O(89^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 15 + 59\cdot 89 + 22\cdot 89^{2} + 65\cdot 89^{3} + 14\cdot 89^{4} + 33\cdot 89^{5} +O(89^{6})\) 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$$1$$()$$5$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$2$
$40$$3$$(1,2,3)$$-1$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$72$$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.