LMFDB - Number field 16.0.24722989956581301387479701814209.4

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648)

gp: K = bnfinit(y^16 - 8*y^15 - 4*y^14 + 168*y^13 - 18*y^12 - 2440*y^11 + 1466*y^10 + 19994*y^9 + 2364*y^8 - 164840*y^7 + 31514*y^6 + 602240*y^5 - 47635*y^4 - 1158746*y^3 - 748608*y^2 + 1464552*y + 8353648, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648)

$ x^{16} - 8 x^{15} - 4 x^{14} + 168 x^{13} - 18 x^{12} - 2440 x^{11} + 1466 x^{10} + 19994 x^{9} + \cdots + 8353648 $ x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648

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:	$24722989956581301387479701814209$ 24722989956581301387479701814209 $\medspace = 17^{14}\cdot 59^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$91.64$	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:	$17^{7/8}59^{1/2}\approx 91.6365672680861$
Ramified primes:	$17$, $59$ 17, 59	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 over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{68}a^{8}-\frac{1}{17}a^{7}-\frac{5}{34}a^{6}+\frac{5}{34}a^{5}-\frac{1}{34}a^{4}-\frac{3}{34}a^{3}-\frac{7}{68}a^{2}+\frac{9}{34}a-\frac{4}{17}$, $\frac{1}{68}a^{9}+\frac{2}{17}a^{7}+\frac{1}{17}a^{6}+\frac{1}{17}a^{5}-\frac{7}{34}a^{4}+\frac{3}{68}a^{3}+\frac{6}{17}a^{2}+\frac{11}{34}a+\frac{1}{17}$, $\frac{1}{68}a^{10}+\frac{1}{34}a^{7}+\frac{4}{17}a^{6}+\frac{2}{17}a^{5}-\frac{15}{68}a^{4}+\frac{1}{17}a^{3}-\frac{6}{17}a^{2}-\frac{1}{17}a-\frac{2}{17}$, $\frac{1}{68}a^{11}-\frac{5}{34}a^{7}-\frac{3}{34}a^{6}-\frac{1}{68}a^{5}+\frac{2}{17}a^{4}+\frac{11}{34}a^{3}-\frac{6}{17}a^{2}-\frac{5}{34}a+\frac{8}{17}$, $\frac{1}{1361528368}a^{12}-\frac{3}{680764184}a^{11}+\frac{55940}{85095523}a^{10}-\frac{4475145}{1361528368}a^{9}+\frac{1507}{85095523}a^{8}+\frac{6688589}{340382092}a^{7}-\frac{321564861}{1361528368}a^{6}+\frac{85747305}{680764184}a^{5}-\frac{1605901}{340382092}a^{4}-\frac{5892075}{1361528368}a^{3}-\frac{60216441}{340382092}a^{2}+\frac{47506667}{170191046}a-\frac{569927}{170191046}$, $\frac{1}{1361528368}a^{13}+\frac{223751}{340382092}a^{11}+\frac{895095}{1361528368}a^{10}-\frac{3402141}{680764184}a^{9}+\frac{859569}{170191046}a^{8}+\frac{79230987}{1361528368}a^{7}-\frac{29012881}{340382092}a^{6}+\frac{27705627}{170191046}a^{5}-\frac{284703411}{1361528368}a^{4}-\frac{48007965}{680764184}a^{3}+\frac{59079923}{340382092}a^{2}-\frac{45900779}{170191046}a+\frac{23318314}{85095523}$, $\frac{1}{1361528368}a^{14}+\frac{6265119}{1361528368}a^{11}+\frac{4028921}{680764184}a^{10}+\frac{217223}{85095523}a^{9}+\frac{3033763}{1361528368}a^{8}-\frac{23047757}{340382092}a^{7}+\frac{17422291}{340382092}a^{6}-\frac{310066099}{1361528368}a^{5}-\frac{69216363}{680764184}a^{4}+\frac{101894075}{340382092}a^{3}-\frac{77175285}{170191046}a^{2}+\frac{9221190}{85095523}a+\frac{42223637}{85095523}$, $\frac{1}{60714634514224}a^{15}+\frac{22289}{60714634514224}a^{14}-\frac{2789}{15178658628556}a^{13}-\frac{2705}{30357317257112}a^{12}-\frac{366452701497}{60714634514224}a^{11}+\frac{107507912861}{30357317257112}a^{10}+\frac{8711369307}{7589329314278}a^{9}+\frac{22998320637}{3195507079696}a^{8}+\frac{576615903997}{3794664657139}a^{7}-\frac{1790296669629}{30357317257112}a^{6}-\frac{12532958676611}{60714634514224}a^{5}+\frac{2814364035929}{30357317257112}a^{4}+\frac{10257132477203}{60714634514224}a^{3}-\frac{6611517114389}{15178658628556}a^{2}+\frac{99639391544}{223215568067}a-\frac{3653440803737}{7589329314278}$ 1, a, a^2, a^3, 1/2*a^4 - 1/2*a, 1/2*a^5 - 1/2*a^2, 1/2*a^6 - 1/2*a^3, 1/2*a^7 - 1/2*a, 1/68*a^8 - 1/17*a^7 - 5/34*a^6 + 5/34*a^5 - 1/34*a^4 - 3/34*a^3 - 7/68*a^2 + 9/34*a - 4/17, 1/68*a^9 + 2/17*a^7 + 1/17*a^6 + 1/17*a^5 - 7/34*a^4 + 3/68*a^3 + 6/17*a^2 + 11/34*a + 1/17, 1/68*a^10 + 1/34*a^7 + 4/17*a^6 + 2/17*a^5 - 15/68*a^4 + 1/17*a^3 - 6/17*a^2 - 1/17*a - 2/17, 1/68*a^11 - 5/34*a^7 - 3/34*a^6 - 1/68*a^5 + 2/17*a^4 + 11/34*a^3 - 6/17*a^2 - 5/34*a + 8/17, 1/1361528368*a^12 - 3/680764184*a^11 + 55940/85095523*a^10 - 4475145/1361528368*a^9 + 1507/85095523*a^8 + 6688589/340382092*a^7 - 321564861/1361528368*a^6 + 85747305/680764184*a^5 - 1605901/340382092*a^4 - 5892075/1361528368*a^3 - 60216441/340382092*a^2 + 47506667/170191046*a - 569927/170191046, 1/1361528368*a^13 + 223751/340382092*a^11 + 895095/1361528368*a^10 - 3402141/680764184*a^9 + 859569/170191046*a^8 + 79230987/1361528368*a^7 - 29012881/340382092*a^6 + 27705627/170191046*a^5 - 284703411/1361528368*a^4 - 48007965/680764184*a^3 + 59079923/340382092*a^2 - 45900779/170191046*a + 23318314/85095523, 1/1361528368*a^14 + 6265119/1361528368*a^11 + 4028921/680764184*a^10 + 217223/85095523*a^9 + 3033763/1361528368*a^8 - 23047757/340382092*a^7 + 17422291/340382092*a^6 - 310066099/1361528368*a^5 - 69216363/680764184*a^4 + 101894075/340382092*a^3 - 77175285/170191046*a^2 + 9221190/85095523*a + 42223637/85095523, 1/60714634514224*a^15 + 22289/60714634514224*a^14 - 2789/15178658628556*a^13 - 2705/30357317257112*a^12 - 366452701497/60714634514224*a^11 + 107507912861/30357317257112*a^10 + 8711369307/7589329314278*a^9 + 22998320637/3195507079696*a^8 + 576615903997/3794664657139*a^7 - 1790296669629/30357317257112*a^6 - 12532958676611/60714634514224*a^5 + 2814364035929/30357317257112*a^4 + 10257132477203/60714634514224*a^3 - 6611517114389/15178658628556*a^2 + 99639391544/223215568067*a - 3653440803737/7589329314278

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

$C_{24}$, which has order $24$ (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:	$ -1 $ -1 (order $2$)	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:	$\frac{1}{4599758}a^{14}-\frac{7}{4599758}a^{13}+\frac{165}{340382092}a^{12}+\frac{1436}{85095523}a^{11}-\frac{5983}{170191046}a^{10}-\frac{5169}{340382092}a^{9}+\frac{41315}{85095523}a^{8}-\frac{272839}{170191046}a^{7}-\frac{224857}{340382092}a^{6}+\frac{619332}{85095523}a^{5}-\frac{204381}{170191046}a^{4}-\frac{3894827}{340382092}a^{3}-\frac{454059}{170191046}a^{2}+\frac{837831}{85095523}a-\frac{8499}{85095523}$, $\frac{18767271}{7589329314278}a^{15}-\frac{962692101}{60714634514224}a^{14}-\frac{1250056099}{30357317257112}a^{13}+\frac{6453031713}{15178658628556}a^{12}+\frac{25389945517}{60714634514224}a^{11}-\frac{93834502937}{15178658628556}a^{10}-\frac{26925322969}{15178658628556}a^{9}+\frac{160553360283}{3195507079696}a^{8}+\frac{1057702814541}{30357317257112}a^{7}-\frac{2218738994643}{7589329314278}a^{6}-\frac{19581670280729}{60714634514224}a^{5}+\frac{8266055560571}{7589329314278}a^{4}+\frac{6851826908341}{7589329314278}a^{3}-\frac{18306771189997}{7589329314278}a^{2}+\frac{5902218073080}{3794664657139}a+\frac{1556774295611}{223215568067}$, $\frac{6497567}{3794664657139}a^{15}-\frac{721960105}{60714634514224}a^{14}-\frac{107242157}{30357317257112}a^{13}+\frac{1284009761}{7589329314278}a^{12}+\frac{519104145}{60714634514224}a^{11}-\frac{21661955485}{15178658628556}a^{10}+\frac{1047997571}{15178658628556}a^{9}+\frac{1577021291}{3195507079696}a^{8}+\frac{34686107543}{820468033976}a^{7}-\frac{253853230769}{15178658628556}a^{6}-\frac{9456645468357}{60714634514224}a^{5}-\frac{1037949396418}{3794664657139}a^{4}+\frac{8551184759425}{7589329314278}a^{3}-\frac{15834927714867}{15178658628556}a^{2}+\frac{1326623377714}{3794664657139}a-\frac{23679064254214}{3794664657139}$, $\frac{43208621}{7589329314278}a^{15}-\frac{2187924971}{60714634514224}a^{14}-\frac{4357727173}{60714634514224}a^{13}+\frac{45864802599}{60714634514224}a^{12}+\frac{69750946313}{60714634514224}a^{11}-\frac{639340710413}{60714634514224}a^{10}-\frac{41669612301}{3571449089072}a^{9}+\frac{279772786237}{3195507079696}a^{8}+\frac{594088122113}{3571449089072}a^{7}-\frac{33702817327715}{60714634514224}a^{6}-\frac{57594977877693}{60714634514224}a^{5}+\frac{94603874278521}{60714634514224}a^{4}+\frac{231724856104561}{60714634514224}a^{3}+\frac{3714001715627}{3794664657139}a^{2}-\frac{28288786181958}{3794664657139}a-\frac{124246909213515}{7589329314278}$, $\frac{63702613}{30357317257112}a^{15}-\frac{140178291}{30357317257112}a^{14}-\frac{984662357}{7589329314278}a^{13}+\frac{16103000717}{30357317257112}a^{12}+\frac{52662912339}{30357317257112}a^{11}-\frac{34483344319}{3794664657139}a^{10}-\frac{510346656313}{30357317257112}a^{9}+\frac{155334982701}{1597753539848}a^{8}+\frac{589023010771}{7589329314278}a^{7}-\frac{10587736672909}{30357317257112}a^{6}-\frac{31563085957967}{30357317257112}a^{5}+\frac{5947087529529}{3794664657139}a^{4}+\frac{13135567800723}{7589329314278}a^{3}+\frac{14353067774094}{3794664657139}a^{2}-\frac{18383789183032}{3794664657139}a-\frac{19194504945027}{3794664657139}$, $\frac{63702613}{30357317257112}a^{15}-\frac{101920113}{3794664657139}a^{14}+\frac{787628863}{30357317257112}a^{13}+\frac{18870977413}{30357317257112}a^{12}-\frac{6346641405}{7589329314278}a^{11}-\frac{290750069639}{30357317257112}a^{10}+\frac{392166433015}{30357317257112}a^{9}+\frac{65143794909}{798876769924}a^{8}-\frac{2207031118643}{30357317257112}a^{7}-\frac{20090572754269}{30357317257112}a^{6}+\frac{2221026240446}{3794664657139}a^{5}+\frac{79751480615107}{30357317257112}a^{4}-\frac{25421615179125}{7589329314278}a^{3}-\frac{94220141892785}{15178658628556}a^{2}+\frac{30330328564032}{3794664657139}a+\frac{15405958491498}{3794664657139}$, $\frac{6497567}{3794664657139}a^{15}-\frac{837455975}{60714634514224}a^{14}+\frac{74248347}{7589329314278}a^{13}+\frac{10912299789}{60714634514224}a^{12}-\frac{13832350231}{60714634514224}a^{11}-\frac{62594998885}{30357317257112}a^{10}+\frac{347726428859}{60714634514224}a^{9}+\frac{22959360097}{3195507079696}a^{8}-\frac{163798779935}{7589329314278}a^{7}-\frac{7662500842989}{60714634514224}a^{6}+\frac{25690080033111}{60714634514224}a^{5}+\frac{166577486877}{30357317257112}a^{4}-\frac{29355963040371}{60714634514224}a^{3}+\frac{3981031364813}{15178658628556}a^{2}-\frac{158342518904}{3794664657139}a+\frac{47140537164031}{7589329314278}$ 1/4599758a^14 - 7/4599758a^13 + 165/340382092a^12 + 1436/85095523a^11 - 5983/170191046a^10 - 5169/340382092a^9 + 41315/85095523a^8 - 272839/170191046a^7 - 224857/340382092a^6 + 619332/85095523a^5 - 204381/170191046a^4 - 3894827/340382092a^3 - 454059/170191046a^2 + 837831/85095523a - 8499/85095523, 18767271/7589329314278a^15 - 962692101/60714634514224a^14 - 1250056099/30357317257112a^13 + 6453031713/15178658628556a^12 + 25389945517/60714634514224a^11 - 93834502937/15178658628556a^10 - 26925322969/15178658628556a^9 + 160553360283/3195507079696a^8 + 1057702814541/30357317257112a^7 - 2218738994643/7589329314278a^6 - 19581670280729/60714634514224a^5 + 8266055560571/7589329314278a^4 + 6851826908341/7589329314278a^3 - 18306771189997/7589329314278a^2 + 5902218073080/3794664657139a + 1556774295611/223215568067, 6497567/3794664657139a^15 - 721960105/60714634514224a^14 - 107242157/30357317257112a^13 + 1284009761/7589329314278a^12 + 519104145/60714634514224a^11 - 21661955485/15178658628556a^10 + 1047997571/15178658628556a^9 + 1577021291/3195507079696a^8 + 34686107543/820468033976a^7 - 253853230769/15178658628556a^6 - 9456645468357/60714634514224a^5 - 1037949396418/3794664657139a^4 + 8551184759425/7589329314278a^3 - 15834927714867/15178658628556a^2 + 1326623377714/3794664657139a - 23679064254214/3794664657139, 43208621/7589329314278a^15 - 2187924971/60714634514224a^14 - 4357727173/60714634514224a^13 + 45864802599/60714634514224a^12 + 69750946313/60714634514224a^11 - 639340710413/60714634514224a^10 - 41669612301/3571449089072a^9 + 279772786237/3195507079696a^8 + 594088122113/3571449089072a^7 - 33702817327715/60714634514224a^6 - 57594977877693/60714634514224a^5 + 94603874278521/60714634514224a^4 + 231724856104561/60714634514224a^3 + 3714001715627/3794664657139a^2 - 28288786181958/3794664657139a - 124246909213515/7589329314278, 63702613/30357317257112a^15 - 140178291/30357317257112a^14 - 984662357/7589329314278a^13 + 16103000717/30357317257112a^12 + 52662912339/30357317257112a^11 - 34483344319/3794664657139a^10 - 510346656313/30357317257112a^9 + 155334982701/1597753539848a^8 + 589023010771/7589329314278a^7 - 10587736672909/30357317257112a^6 - 31563085957967/30357317257112a^5 + 5947087529529/3794664657139a^4 + 13135567800723/7589329314278a^3 + 14353067774094/3794664657139a^2 - 18383789183032/3794664657139a - 19194504945027/3794664657139, 63702613/30357317257112a^15 - 101920113/3794664657139a^14 + 787628863/30357317257112a^13 + 18870977413/30357317257112a^12 - 6346641405/7589329314278a^11 - 290750069639/30357317257112a^10 + 392166433015/30357317257112a^9 + 65143794909/798876769924a^8 - 2207031118643/30357317257112a^7 - 20090572754269/30357317257112a^6 + 2221026240446/3794664657139a^5 + 79751480615107/30357317257112a^4 - 25421615179125/7589329314278a^3 - 94220141892785/15178658628556a^2 + 30330328564032/3794664657139a + 15405958491498/3794664657139, 6497567/3794664657139a^15 - 837455975/60714634514224a^14 + 74248347/7589329314278a^13 + 10912299789/60714634514224a^12 - 13832350231/60714634514224a^11 - 62594998885/30357317257112a^10 + 347726428859/60714634514224a^9 + 22959360097/3195507079696a^8 - 163798779935/7589329314278a^7 - 7662500842989/60714634514224a^6 + 25690080033111/60714634514224a^5 + 166577486877/30357317257112a^4 - 29355963040371/60714634514224a^3 + 3981031364813/15178658628556a^2 - 158342518904/3794664657139a + 47140537164031/7589329314278 (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:	$ 335717402.221 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 335717402.221 \cdot 24}{2\cdot\sqrt{24722989956581301387479701814209}}\cr\approx \mathstrut & 1.96808363712 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648)

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 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648, 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 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648);

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 - 8*x^15 - 4*x^14 + 168*x^13 - 18*x^12 - 2440*x^11 + 1466*x^10 + 19994*x^9 + 2364*x^8 - 164840*x^7 + 31514*x^6 + 602240*x^5 - 47635*x^4 - 1158746*x^3 - 748608*x^2 + 1464552*x + 8353648);

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

$\SD_{16}$ (as 16T12):

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)

A solvable group of order 16

The 7 conjugacy class representatives for $QD_{16}$

Character table for $QD_{16}$

Intermediate fields

$\Q(\sqrt{-59}) $, $\Q(\sqrt{17}) $, $\Q(\sqrt{-1003}) $, $\Q(\sqrt{17}, \sqrt{-59})$, 4.2.289867.1 x2, 4.0.17102153.1 x2, 8.0.292483637235409.1, 8.2.84274946322067.1 x4

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]

Sibling fields

Degree 8 sibling:	8.2.84274946322067.1
Minimal sibling:	8.2.84274946322067.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.2.0.1}{2} }^{8}$	${\href{/padicField/3.8.0.1}{8} }^{2}$	${\href{/padicField/5.8.0.1}{8} }^{2}$	${\href{/padicField/7.8.0.1}{8} }^{2}$	${\href{/padicField/11.4.0.1}{4} }^{4}$	${\href{/padicField/13.2.0.1}{2} }^{8}$	R	${\href{/padicField/19.2.0.1}{2} }^{8}$	${\href{/padicField/23.4.0.1}{4} }^{4}$	${\href{/padicField/29.8.0.1}{8} }^{2}$	${\href{/padicField/31.4.0.1}{4} }^{4}$	${\href{/padicField/37.4.0.1}{4} }^{4}$	${\href{/padicField/41.8.0.1}{8} }^{2}$	${\href{/padicField/43.2.0.1}{2} }^{8}$	${\href{/padicField/47.2.0.1}{2} }^{8}$	${\href{/padicField/53.1.0.1}{1} }^{16}$	R

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
$17$ 17	17.8.7.2	$x^{8} + 136$	$8$	$1$	$7$	$C_8$	$[\ ]_{8}$
$17$ 17	17.8.7.2	$x^{8} + 136$	$8$	$1$	$7$	$C_8$	$[\ ]_{8}$
$59$ 59	59.4.2.1	$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	59.4.2.1	$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	59.4.2.1	$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	59.4.2.1	$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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