Properties

Label 10.657...049.30t88.a.a
Dimension $10$
Group $A_6$
Conductor $6.580\times 10^{18}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $10$
Group: $A_6$
Conductor: \(6579780298306547049\)\(\medspace = 3^{18} \cdot 19^{8} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.95004009.3
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 6.2.95004009.3

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} - 3x^{4} + 8x^{3} + 9x^{2} - 9x + 6 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 11 + 42\cdot 311 + 169\cdot 311^{2} + 275\cdot 311^{3} + 188\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 178 + 66\cdot 311 + 83\cdot 311^{2} + 306\cdot 311^{3} + 283\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 224 + 261\cdot 311 + 143\cdot 311^{2} + 74\cdot 311^{3} + 94\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 261 + 310\cdot 311 + 19\cdot 311^{2} + 227\cdot 311^{3} + 110\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 271 + 309\cdot 311 + 283\cdot 311^{2} + 257\cdot 311^{3} + 5\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 302 + 252\cdot 311 + 232\cdot 311^{2} + 102\cdot 311^{3} + 249\cdot 311^{4} +O(311^{5})\) 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.