LMFDB - Number field 16.0.125439056256992431640625.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616)

gp: K = bnfinit(y^16 - y^15 + 6*y^14 - 17*y^13 + 17*y^12 + 49*y^11 - 78*y^10 + 833*y^9 - 2129*y^8 + 4998*y^7 - 2808*y^6 + 10584*y^5 + 22032*y^4 - 132192*y^3 + 279936*y^2 - 279936*y + 1679616, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616)

$ x^{16} - x^{15} + 6 x^{14} - 17 x^{13} + 17 x^{12} + 49 x^{11} - 78 x^{10} + 833 x^{9} - 2129 x^{8} + \cdots + 1679616 $ x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$16$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 8]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$125439056256992431640625$ 125439056256992431640625 $\medspace = 3^{8}\cdot 5^{12}\cdot 23^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$27.77$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$3^{1/2}5^{3/4}23^{1/2}\approx 27.77487087706129$
Ramified primes:	$3$, $5$, $23$ 3, 5, 23	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$16$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$345=3\cdot 5\cdot 23$
Dirichlet character group:	$\lbrace$$\chi_{345}(1,·)$, $\chi_{345}(323,·)$, $\chi_{345}(68,·)$, $\chi_{345}(137,·)$, $\chi_{345}(139,·)$, $\chi_{345}(206,·)$, $\chi_{345}(208,·)$, $\chi_{345}(277,·)$, $\chi_{345}(22,·)$, $\chi_{345}(344,·)$, $\chi_{345}(91,·)$, $\chi_{345}(229,·)$, $\chi_{345}(298,·)$, $\chi_{345}(47,·)$, $\chi_{345}(116,·)$, $\chi_{345}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{128}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{8}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{2}+\frac{1}{6}a$, $\frac{1}{684}a^{10}-\frac{1}{36}a^{9}-\frac{1}{3}a^{8}+\frac{1}{36}a^{7}+\frac{17}{36}a^{6}-\frac{257}{684}a^{5}+\frac{1}{3}a^{4}-\frac{13}{36}a^{3}-\frac{5}{36}a^{2}+\frac{1}{3}a+\frac{7}{19}$, $\frac{1}{4104}a^{11}-\frac{1}{4104}a^{10}+\frac{1}{36}a^{9}+\frac{73}{216}a^{8}+\frac{35}{216}a^{7}+\frac{1453}{4104}a^{6}+\frac{293}{684}a^{5}+\frac{95}{216}a^{4}+\frac{13}{216}a^{3}+\frac{5}{36}a^{2}+\frac{15}{38}a+\frac{2}{19}$, $\frac{1}{123120}a^{12}+\frac{1}{24624}a^{11}+\frac{1}{6480}a^{9}+\frac{613}{1296}a^{8}-\frac{6709}{24624}a^{7}+\frac{1}{570}a^{6}-\frac{269}{1296}a^{5}-\frac{229}{1296}a^{4}+\frac{1}{5}a^{3}-\frac{29}{684}a^{2}+\frac{17}{114}a-\frac{1}{5}$, $\frac{1}{2219853600}a^{13}+\frac{7421}{2219853600}a^{12}+\frac{1261}{12332520}a^{11}-\frac{189881}{2219853600}a^{10}-\frac{6081799}{116834400}a^{9}+\frac{209839703}{443970720}a^{8}-\frac{24907349}{61662600}a^{7}+\frac{274669241}{2219853600}a^{6}-\frac{46734367}{443970720}a^{5}-\frac{1107749}{3245400}a^{4}+\frac{901616}{2569275}a^{3}-\frac{4643}{114190}a^{2}+\frac{77621}{285475}a-\frac{83614}{285475}$, $\frac{1}{13319121600}a^{14}-\frac{1}{13319121600}a^{13}-\frac{737}{2219853600}a^{12}+\frac{1149859}{13319121600}a^{11}-\frac{1171999}{13319121600}a^{10}-\frac{458842163}{13319121600}a^{9}-\frac{159295099}{2219853600}a^{8}-\frac{4019426851}{13319121600}a^{7}-\frac{2348716577}{13319121600}a^{6}+\frac{750753959}{2219853600}a^{5}-\frac{59664203}{123325200}a^{4}+\frac{1024343}{6851400}a^{3}+\frac{122706}{285475}a^{2}-\frac{413651}{1712850}a-\frac{25622}{285475}$, $\frac{1}{79914729600}a^{15}-\frac{1}{79914729600}a^{14}+\frac{1}{13319121600}a^{13}+\frac{13573}{4206038400}a^{12}-\frac{114661}{4206038400}a^{11}+\frac{5015569}{79914729600}a^{10}-\frac{820750117}{13319121600}a^{9}-\frac{3458710831}{79914729600}a^{8}+\frac{414681349}{4206038400}a^{7}+\frac{11167403}{701006400}a^{6}+\frac{126726281}{369975600}a^{5}+\frac{179200459}{369975600}a^{4}-\frac{22444417}{61662600}a^{3}+\frac{63389}{135225}a^{2}-\frac{20998}{45075}a+\frac{91743}{285475}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/6*a^9 - 1/6*a^8 + 1/6*a^6 - 1/6*a^5 + 1/6*a^4 - 1/6*a^2 + 1/6*a, 1/684*a^10 - 1/36*a^9 - 1/3*a^8 + 1/36*a^7 + 17/36*a^6 - 257/684*a^5 + 1/3*a^4 - 13/36*a^3 - 5/36*a^2 + 1/3*a + 7/19, 1/4104*a^11 - 1/4104*a^10 + 1/36*a^9 + 73/216*a^8 + 35/216*a^7 + 1453/4104*a^6 + 293/684*a^5 + 95/216*a^4 + 13/216*a^3 + 5/36*a^2 + 15/38*a + 2/19, 1/123120*a^12 + 1/24624*a^11 + 1/6480*a^9 + 613/1296*a^8 - 6709/24624*a^7 + 1/570*a^6 - 269/1296*a^5 - 229/1296*a^4 + 1/5*a^3 - 29/684*a^2 + 17/114*a - 1/5, 1/2219853600*a^13 + 7421/2219853600*a^12 + 1261/12332520*a^11 - 189881/2219853600*a^10 - 6081799/116834400*a^9 + 209839703/443970720*a^8 - 24907349/61662600*a^7 + 274669241/2219853600*a^6 - 46734367/443970720*a^5 - 1107749/3245400*a^4 + 901616/2569275*a^3 - 4643/114190*a^2 + 77621/285475*a - 83614/285475, 1/13319121600*a^14 - 1/13319121600*a^13 - 737/2219853600*a^12 + 1149859/13319121600*a^11 - 1171999/13319121600*a^10 - 458842163/13319121600*a^9 - 159295099/2219853600*a^8 - 4019426851/13319121600*a^7 - 2348716577/13319121600*a^6 + 750753959/2219853600*a^5 - 59664203/123325200*a^4 + 1024343/6851400*a^3 + 122706/285475*a^2 - 413651/1712850*a - 25622/285475, 1/79914729600*a^15 - 1/79914729600*a^14 + 1/13319121600*a^13 + 13573/4206038400*a^12 - 114661/4206038400*a^11 + 5015569/79914729600*a^10 - 820750117/13319121600*a^9 - 3458710831/79914729600*a^8 + 414681349/4206038400*a^7 + 11167403/701006400*a^6 + 126726281/369975600*a^5 + 179200459/369975600*a^4 - 22444417/61662600*a^3 + 63389/135225*a^2 - 20998/45075*a + 91743/285475

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{15}$, which has order $15$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$7$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{1}{2055420} a^{15} + \frac{11}{10277100} a^{14} - \frac{11}{1712850} a^{12} + \frac{29}{10277100} a^{11} - \frac{151}{2055420} a^{10} + \frac{36}{285475} a^{9} - \frac{151}{540900} a^{8} - \frac{73}{342570} a^{7} - \frac{216}{285475} a^{6} - \frac{906}{285475} a^{5} + \frac{612}{57095} a^{4} - \frac{216}{15025} a^{3} + \frac{1609901}{10277100} a^{2} - \frac{7776}{57095} a - \frac{46656}{285475} $ -(1)/(2055420)a^(15) + (11)/(10277100)a^(14) - (11)/(1712850)a^(12) + (29)/(10277100)a^(11) - (151)/(2055420)a^(10) + (36)/(285475)a^(9) - (151)/(540900)a^(8) - (73)/(342570)a^(7) - (216)/(285475)a^(6) - (906)/(285475)a^(5) + (612)/(57095)a^(4) - (216)/(15025)a^(3) + (1609901)/(10277100)a^(2) - (7776)/(57095)a - (46656)/(285475) (order $30$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{2567}{2663824320}a^{15}-\frac{2567}{443970720}a^{14}+\frac{43639}{2663824320}a^{13}-\frac{43639}{2663824320}a^{12}+\frac{21607}{443970720}a^{11}+\frac{33371}{443970720}a^{10}-\frac{2138311}{2663824320}a^{9}+\frac{5465143}{2663824320}a^{8}-\frac{2138311}{443970720}a^{7}+\frac{22410431}{2663824320}a^{6}-\frac{125783}{12332520}a^{5}-\frac{43639}{2055420}a^{4}+\frac{43639}{342570}a^{3}-\frac{15402}{57095}a^{2}+\frac{37434}{57095}a-\frac{35317}{57095}$, $\frac{4379}{15982945920}a^{15}-\frac{48169}{79914729600}a^{14}-\frac{146711}{79914729600}a^{12}-\frac{126991}{79914729600}a^{11}+\frac{661229}{15982945920}a^{10}-\frac{4379}{61662600}a^{9}+\frac{661229}{4206038400}a^{8}-\frac{34607773}{15982945920}a^{7}+\frac{4379}{10277100}a^{6}+\frac{661229}{369975600}a^{5}-\frac{74443}{12332520}a^{4}+\frac{4379}{540900}a^{3}-\frac{6264}{285475}a^{2}+\frac{4379}{57095}a-\frac{259201}{285475}$, $\frac{4379}{13319121600}a^{15}+\frac{5059}{2219853600}a^{14}-\frac{11023}{2663824320}a^{13}+\frac{180611}{13319121600}a^{12}-\frac{22499}{2219853600}a^{11}+\frac{659}{12332520}a^{10}+\frac{7612259}{13319121600}a^{9}-\frac{9745691}{13319121600}a^{8}+\frac{322643}{88794144}a^{7}-\frac{51585979}{13319121600}a^{6}+\frac{43438321}{2219853600}a^{5}+\frac{27871}{2055420}a^{4}-\frac{1436161}{61662600}a^{3}+\frac{1162459}{10277100}a^{2}-\frac{15402}{57095}a+\frac{92412}{57095}$, $\frac{629681}{79914729600}a^{15}-\frac{4273}{841207680}a^{14}+\frac{17411}{665956080}a^{13}-\frac{453581}{15982945920}a^{12}+\frac{2219987}{15982945920}a^{11}+\frac{11863171}{15982945920}a^{10}-\frac{10841}{8762580}a^{9}+\frac{47373053}{15982945920}a^{8}-\frac{124037171}{15982945920}a^{7}+\frac{18088801}{665956080}a^{6}+\frac{7515641}{221985360}a^{5}-\frac{117431}{3894480}a^{4}+\frac{2445491}{12332520}a^{3}-\frac{268391}{1027710}a^{2}+\frac{749849}{342570}a-\frac{9359}{285475}$, $\frac{118171}{15982945920}a^{15}+\frac{417997}{79914729600}a^{14}-\frac{202427}{13319121600}a^{13}+\frac{3088841}{79914729600}a^{12}-\frac{13544477}{79914729600}a^{11}+\frac{67640387}{79914729600}a^{10}+\frac{3784919}{13319121600}a^{9}-\frac{87659993}{79914729600}a^{8}+\frac{419273213}{79914729600}a^{7}-\frac{175541899}{13319121600}a^{6}+\frac{80237179}{1109926800}a^{5}-\frac{7195693}{369975600}a^{4}-\frac{4049981}{61662600}a^{3}-\frac{1063529}{5138550}a^{2}-\frac{687376}{856425}a+\frac{1343341}{285475}$, $\frac{7411}{9989341200}a^{15}+\frac{46069}{4994670600}a^{14}+\frac{7411}{1664890200}a^{13}-\frac{125987}{9989341200}a^{12}+\frac{125987}{9989341200}a^{11}+\frac{363139}{9989341200}a^{10}+\frac{4269739}{3329780400}a^{9}+\frac{6173363}{9989341200}a^{8}-\frac{15778019}{9989341200}a^{7}+\frac{6173363}{1664890200}a^{6}-\frac{96343}{46246950}a^{5}+\frac{19357487}{369975600}a^{4}+\frac{125987}{7707825}a^{3}-\frac{251974}{2569275}a^{2}+\frac{59288}{285475}a-\frac{59288}{285475}$, $\frac{6787}{208111275}a^{15}-\frac{933887}{13319121600}a^{14}+\frac{653323}{13319121600}a^{13}-\frac{167087}{1331912160}a^{12}+\frac{4352047}{13319121600}a^{11}+\frac{1438463}{701006400}a^{10}-\frac{17158199}{2663824320}a^{9}+\frac{113006659}{6659560800}a^{8}-\frac{506125567}{13319121600}a^{7}+\frac{223669687}{2663824320}a^{6}+\frac{5079053}{116834400}a^{5}-\frac{10808893}{61662600}a^{4}+\frac{205003}{616626}a^{3}-\frac{30412087}{10277100}a^{2}+\frac{16639043}{1712850}a-\frac{606807}{285475}$ 2567/2663824320a^15 - 2567/443970720a^14 + 43639/2663824320a^13 - 43639/2663824320a^12 + 21607/443970720a^11 + 33371/443970720a^10 - 2138311/2663824320a^9 + 5465143/2663824320a^8 - 2138311/443970720a^7 + 22410431/2663824320a^6 - 125783/12332520a^5 - 43639/2055420a^4 + 43639/342570a^3 - 15402/57095a^2 + 37434/57095a - 35317/57095, 4379/15982945920a^15 - 48169/79914729600a^14 - 146711/79914729600a^12 - 126991/79914729600a^11 + 661229/15982945920a^10 - 4379/61662600a^9 + 661229/4206038400a^8 - 34607773/15982945920a^7 + 4379/10277100a^6 + 661229/369975600a^5 - 74443/12332520a^4 + 4379/540900a^3 - 6264/285475a^2 + 4379/57095a - 259201/285475, 4379/13319121600a^15 + 5059/2219853600a^14 - 11023/2663824320a^13 + 180611/13319121600a^12 - 22499/2219853600a^11 + 659/12332520a^10 + 7612259/13319121600a^9 - 9745691/13319121600a^8 + 322643/88794144a^7 - 51585979/13319121600a^6 + 43438321/2219853600a^5 + 27871/2055420a^4 - 1436161/61662600a^3 + 1162459/10277100a^2 - 15402/57095a + 92412/57095, 629681/79914729600a^15 - 4273/841207680a^14 + 17411/665956080a^13 - 453581/15982945920a^12 + 2219987/15982945920a^11 + 11863171/15982945920a^10 - 10841/8762580a^9 + 47373053/15982945920a^8 - 124037171/15982945920a^7 + 18088801/665956080a^6 + 7515641/221985360a^5 - 117431/3894480a^4 + 2445491/12332520a^3 - 268391/1027710a^2 + 749849/342570a - 9359/285475, 118171/15982945920a^15 + 417997/79914729600a^14 - 202427/13319121600a^13 + 3088841/79914729600a^12 - 13544477/79914729600a^11 + 67640387/79914729600a^10 + 3784919/13319121600a^9 - 87659993/79914729600a^8 + 419273213/79914729600a^7 - 175541899/13319121600a^6 + 80237179/1109926800a^5 - 7195693/369975600a^4 - 4049981/61662600a^3 - 1063529/5138550a^2 - 687376/856425a + 1343341/285475, 7411/9989341200a^15 + 46069/4994670600a^14 + 7411/1664890200a^13 - 125987/9989341200a^12 + 125987/9989341200a^11 + 363139/9989341200a^10 + 4269739/3329780400a^9 + 6173363/9989341200a^8 - 15778019/9989341200a^7 + 6173363/1664890200a^6 - 96343/46246950a^5 + 19357487/369975600a^4 + 125987/7707825a^3 - 251974/2569275a^2 + 59288/285475a - 59288/285475, 6787/208111275a^15 - 933887/13319121600a^14 + 653323/13319121600a^13 - 167087/1331912160a^12 + 4352047/13319121600a^11 + 1438463/701006400a^10 - 17158199/2663824320a^9 + 113006659/6659560800a^8 - 506125567/13319121600a^7 + 223669687/2663824320a^6 + 5079053/116834400a^5 - 10808893/61662600a^4 + 205003/616626a^3 - 30412087/10277100a^2 + 16639043/1712850a - 606807/285475 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 93215.7577208 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 93215.7577208 \cdot 15}{30\cdot\sqrt{125439056256992431640625}}\cr\approx \mathstrut & 0.319655290293 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^2\times C_4$ (as 16T2):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

An abelian group of order 16

The 16 conjugacy class representatives for $C_4\times C_2^2$

Character table for $C_4\times C_2^2$

Intermediate fields

$\Q(\sqrt{-3}) $, $\Q(\sqrt{345}) $, $\Q(\sqrt{-115}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{-15}) $, $\Q(\sqrt{69}) $, $\Q(\sqrt{-23}) $, $\Q(\sqrt{-3}, \sqrt{-115})$, $\Q(\sqrt{-3}, \sqrt{5})$, $\Q(\sqrt{-3}, \sqrt{-23})$, $\Q(\sqrt{5}, \sqrt{69})$, $\Q(\sqrt{-15}, \sqrt{-23})$, $\Q(\sqrt{5}, \sqrt{-23})$, $\Q(\sqrt{-15}, \sqrt{69})$, 4.4.66125.1, 4.0.595125.1, $\Q(\zeta_{15})^+$, $\Q(\zeta_{5})$, 8.0.14166950625.1, 8.0.354173765625.2, $\Q(\zeta_{15})$, 8.8.354173765625.1, 8.0.354173765625.3, 8.0.4372515625.1, 8.0.354173765625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.4.0.1}{4} }^{4}$	R	R	${\href{/padicField/7.4.0.1}{4} }^{4}$	${\href{/padicField/11.2.0.1}{2} }^{8}$	${\href{/padicField/13.4.0.1}{4} }^{4}$	${\href{/padicField/17.4.0.1}{4} }^{4}$	${\href{/padicField/19.2.0.1}{2} }^{8}$	R	${\href{/padicField/29.2.0.1}{2} }^{8}$	${\href{/padicField/31.1.0.1}{1} }^{16}$	${\href{/padicField/37.4.0.1}{4} }^{4}$	${\href{/padicField/41.2.0.1}{2} }^{8}$	${\href{/padicField/43.4.0.1}{4} }^{4}$	${\href{/padicField/47.4.0.1}{4} }^{4}$	${\href{/padicField/53.4.0.1}{4} }^{4}$	${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$3$ 3	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$5$ 5	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$5$ 5	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$23$ 23	23.8.4.1	$x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$23$ 23	23.8.4.1	$x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$

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