Properties

Label 3.2e2_5e2_19e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 2^{2} \cdot 5^{2} \cdot 19^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$36100= 2^{2} \cdot 5^{2} \cdot 19^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 11 x^{3} + 6 x^{2} + 64 x - 74 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 557 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 14 + 251\cdot 557 + 288\cdot 557^{2} + 375\cdot 557^{3} + 218\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 235 + 435\cdot 557 + 247\cdot 557^{2} + 321\cdot 557^{3} + 239\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 375 + 224\cdot 557 + 116\cdot 557^{2} + 75\cdot 557^{3} + 92\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 506 + 262\cdot 557 + 255\cdot 557^{2} + 489\cdot 557^{3} + 304\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 542 + 496\cdot 557 + 205\cdot 557^{2} + 409\cdot 557^{3} + 258\cdot 557^{4} +O\left(557^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$3$
$15$$2$$(1,2)(3,4)$$-1$
$20$$3$$(1,2,3)$$0$
$12$$5$$(1,2,3,4,5)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.