Properties

Label 3.261121.12t33.a
Dimension $3$
Group $A_5$
Conductor $261121$
Indicator $1$

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:\(261121\)\(\medspace = 7^{2} \cdot 73^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.261121.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Projective image: $A_5$
Projective field: Galois closure of 5.1.261121.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ a + 3 + \left(22 a + 14\right)\cdot 23 + \left(22 a + 8\right)\cdot 23^{2} + \left(4 a + 18\right)\cdot 23^{3} + \left(12 a + 10\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 21 + \left(5 a + 4\right)\cdot 23 + \left(18 a + 20\right)\cdot 23^{2} + \left(6 a + 20\right)\cdot 23^{3} + \left(a + 2\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 + 2\cdot 23 + 2\cdot 23^{2} + 8\cdot 23^{3} + 3\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 a + 5 + 11\cdot 23 + 9\cdot 23^{2} + \left(18 a + 5\right)\cdot 23^{3} + \left(10 a + 7\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 21 a + 2 + \left(17 a + 13\right)\cdot 23 + \left(4 a + 5\right)\cdot 23^{2} + \left(16 a + 16\right)\cdot 23^{3} + \left(21 a + 21\right)\cdot 23^{4} +O\left(23^{ 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 values
$c1$ $c2$
$1$ $1$ $()$ $3$ $3$
$15$ $2$ $(1,2)(3,4)$ $-1$ $-1$
$20$ $3$ $(1,2,3)$ $0$ $0$
$12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.