Properties

Label 6.132...149.20t30.a.a
Dimension $6$
Group $S_5$
Conductor $1.324\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_5$
Conductor: \(13236200869377149\)\(\medspace = 236549^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.236549.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: even
Determinant: 1.236549.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.236549.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} - 6x^{3} + 7x^{2} + 2x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 461 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 98 + 99\cdot 461 + 277\cdot 461^{2} + 443\cdot 461^{3} + 317\cdot 461^{4} +O(461^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 203 + 279\cdot 461 + 175\cdot 461^{2} + 315\cdot 461^{3} + 430\cdot 461^{4} +O(461^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 247 + 346\cdot 461 + 218\cdot 461^{2} + 382\cdot 461^{3} + 138\cdot 461^{4} +O(461^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 386 + 441\cdot 461 + 9\cdot 461^{2} + 144\cdot 461^{3} + 38\cdot 461^{4} +O(461^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 450 + 215\cdot 461 + 240\cdot 461^{2} + 97\cdot 461^{3} + 457\cdot 461^{4} +O(461^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.