LMFDB - Artin representation 2.471.4t3.c.a

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$471$$\medspace = 3 \cdot 157 $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 4.2.73947.1
Galois orbit size:	$1$
Smallest permutation container:	$D_{4}$
Parity:	odd
Determinant:	1.471.2t1.a.a
Projective image:	$C_2^2$
Projective field:	Galois closure of $\Q(\sqrt{-3}, \sqrt{157})$

$f(x)$

$=$

$ x^{4} - 7x^{2} - 27 $

x^4 - 7*x^2 - 27

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 2 + 4\cdot 13 + 7\cdot 13^{4} + 6\cdot 13^{5} + 5\cdot 13^{6} +O(13^{7})$ 2 + 413 + 713^4 + 613^5 + 513^6+O(13^7)
$r_{ 2 }$	$=$	$ 4 + 6\cdot 13 + 13^{2} + 6\cdot 13^{3} + 3\cdot 13^{4} + 12\cdot 13^{5} + 5\cdot 13^{6} +O(13^{7})$ 4 + 613 + 13^2 + 613^3 + 313^4 + 1213^5 + 5*13^6+O(13^7)
$r_{ 3 }$	$=$	$ 9 + 6\cdot 13 + 11\cdot 13^{2} + 6\cdot 13^{3} + 9\cdot 13^{4} + 7\cdot 13^{6} +O(13^{7})$ 9 + 613 + 1113^2 + 613^3 + 913^4 + 7*13^6+O(13^7)
$r_{ 4 }$	$=$	$ 11 + 8\cdot 13 + 12\cdot 13^{2} + 12\cdot 13^{3} + 5\cdot 13^{4} + 6\cdot 13^{5} + 7\cdot 13^{6} +O(13^{7})$ 11 + 813 + 1213^2 + 1213^3 + 513^4 + 613^5 + 713^6+O(13^7)

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$r_{ 1 }$	$=$	\( 2 + 4\cdot 13 + 7\cdot 13^{4} + 6\cdot 13^{5} + 5\cdot 13^{6} +O(13^{7})\) 2 + 413 + 713^4 + 613^5 + 513^6+O(13^7)
$r_{ 2 }$	$=$	\( 4 + 6\cdot 13 + 13^{2} + 6\cdot 13^{3} + 3\cdot 13^{4} + 12\cdot 13^{5} + 5\cdot 13^{6} +O(13^{7})\) 4 + 613 + 13^2 + 613^3 + 313^4 + 1213^5 + 5*13^6+O(13^7)
$r_{ 3 }$	$=$	\( 9 + 6\cdot 13 + 11\cdot 13^{2} + 6\cdot 13^{3} + 9\cdot 13^{4} + 7\cdot 13^{6} +O(13^{7})\) 9 + 613 + 1113^2 + 613^3 + 913^4 + 7*13^6+O(13^7)
$r_{ 4 }$	$=$	\( 11 + 8\cdot 13 + 12\cdot 13^{2} + 12\cdot 13^{3} + 5\cdot 13^{4} + 6\cdot 13^{5} + 7\cdot 13^{6} +O(13^{7})\) 11 + 813 + 1213^2 + 1213^3 + 513^4 + 613^5 + 713^6+O(13^7)