Properties

Label 5.7102225.10t13.a.a
Dimension $5$
Group $\PGL(2,5)$
Conductor $7102225$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $\PGL(2,5)$
Conductor: \(7102225\)\(\medspace = 5^{2} \cdot 13^{2} \cdot 41^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.18927429625.2
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_5$
Projective stem field: 6.2.18927429625.2

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 17 x^{4} + 117 x^{3} - 188 x^{2} + 23 x + 594\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \(x^{2} + 49 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 16 + 36\cdot 53 + 29\cdot 53^{2} + 11\cdot 53^{3} + 11\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 19 a + 23 + \left(40 a + 13\right)\cdot 53 + \left(14 a + 30\right)\cdot 53^{2} + \left(47 a + 43\right)\cdot 53^{3} + \left(24 a + 51\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 34 a + 46 + \left(12 a + 49\right)\cdot 53 + \left(38 a + 48\right)\cdot 53^{2} + \left(5 a + 5\right)\cdot 53^{3} + \left(28 a + 51\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 39 a + \left(32 a + 18\right)\cdot 53 + \left(17 a + 40\right)\cdot 53^{2} + \left(15 a + 6\right)\cdot 53^{3} + \left(49 a + 48\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 14 a + 50 + \left(20 a + 3\right)\cdot 53 + \left(35 a + 25\right)\cdot 53^{2} + \left(37 a + 50\right)\cdot 53^{3} + \left(3 a + 17\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 25 + 37\cdot 53 + 37\cdot 53^{2} + 40\cdot 53^{3} + 31\cdot 53^{4} +O(53^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$10$$2$$(1,3)(2,4)(5,6)$$1$
$15$$2$$(1,5)(2,4)$$1$
$20$$3$$(1,3,4)(2,5,6)$$-1$
$30$$4$$(1,4,5,2)$$-1$
$24$$5$$(1,6,4,3,5)$$0$
$20$$6$$(1,6,3,2,4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.