Properties

Label 3.2e2_3e4_5e4.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 2^{2} \cdot 3^{4} \cdot 5^{4}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$202500= 2^{2} \cdot 3^{4} \cdot 5^{4} $
Artin number field: Splitting field of $f= x^{5} - 15 x^{3} - 65 x^{2} - 90 x - 57 $ 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 $\Q_{ 487 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 128 + 250\cdot 487 + 389\cdot 487^{2} + 431\cdot 487^{3} + 390\cdot 487^{4} +O\left(487^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 186 + 407\cdot 487 + 388\cdot 487^{2} + 261\cdot 487^{3} + 146\cdot 487^{4} +O\left(487^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 291 + 264\cdot 487 + 269\cdot 487^{2} + 279\cdot 487^{3} + 412\cdot 487^{4} +O\left(487^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 421 + 213\cdot 487 + 229\cdot 487^{2} + 153\cdot 487^{3} + 76\cdot 487^{4} +O\left(487^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 435 + 324\cdot 487 + 183\cdot 487^{2} + 334\cdot 487^{3} + 434\cdot 487^{4} +O\left(487^{ 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.