Properties

Label 2.11025.8t5.a.a
Dimension $2$
Group $Q_8$
Conductor $11025$
Root number $1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(11025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Frobenius-Schur indicator: $-1$
Root number: $1$
Artin number field: Galois closure of 8.8.1340095640625.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{21})\)

Defining polynomial

$f(x)$$=$$ x^{8} - x^{7} - 34 x^{6} + 29 x^{5} + 361 x^{4} - 305 x^{3} - 1090 x^{2} + 1345 x - 395 $.

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 42\cdot 79 + 27\cdot 79^{2} + 30\cdot 79^{3} + 73\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 + 63\cdot 79 + 39\cdot 79^{2} + 76\cdot 79^{3} + 39\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 25 + 29\cdot 79 + 38\cdot 79^{2} + 5\cdot 79^{3} + 26\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 45 + 41\cdot 79 + 21\cdot 79^{2} + 49\cdot 79^{3} + 25\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 49 + 16\cdot 79 + 55\cdot 79^{2} + 15\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 53 + 79 + 12\cdot 79^{2} + 68\cdot 79^{3} + 15\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 66 + 72\cdot 79 + 42\cdot 79^{2} + 11\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 72 + 48\cdot 79 + 78\cdot 79^{2} + 58\cdot 79^{3} + 40\cdot 79^{4} +O\left(79^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,5,2,6)(3,4,7,8)$
$(1,2)(3,7)(4,8)(5,6)$
$(1,8,2,4)(3,5,7,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,7)(4,8)(5,6)$$-2$
$2$$4$$(1,8,2,4)(3,5,7,6)$$0$
$2$$4$$(1,5,2,6)(3,4,7,8)$$0$
$2$$4$$(1,3,2,7)(4,5,8,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.