LMFDB - Number field 16.14.210271325920312500000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 24*x^14 + 46*x^13 + 173*x^12 - 506*x^11 - 236*x^10 + 1853*x^9 - 1192*x^8 - 1736*x^7 + 2296*x^6 - 177*x^5 - 847*x^4 + 373*x^3 - 6*x^2 - 17*x + 1)

gp: K = bnfinit(y^16 - y^15 - 24*y^14 + 46*y^13 + 173*y^12 - 506*y^11 - 236*y^10 + 1853*y^9 - 1192*y^8 - 1736*y^7 + 2296*y^6 - 177*y^5 - 847*y^4 + 373*y^3 - 6*y^2 - 17*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 24*x^14 + 46*x^13 + 173*x^12 - 506*x^11 - 236*x^10 + 1853*x^9 - 1192*x^8 - 1736*x^7 + 2296*x^6 - 177*x^5 - 847*x^4 + 373*x^3 - 6*x^2 - 17*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 24*x^14 + 46*x^13 + 173*x^12 - 506*x^11 - 236*x^10 + 1853*x^9 - 1192*x^8 - 1736*x^7 + 2296*x^6 - 177*x^5 - 847*x^4 + 373*x^3 - 6*x^2 - 17*x + 1)

$ x^{16} - x^{15} - 24 x^{14} + 46 x^{13} + 173 x^{12} - 506 x^{11} - 236 x^{10} + 1853 x^{9} - 1192 x^{8} + \cdots + 1 $ x^16 - x^15 - 24*x^14 + 46*x^13 + 173*x^12 - 506*x^11 - 236*x^10 + 1853*x^9 - 1192*x^8 - 1736*x^7 + 2296*x^6 - 177*x^5 - 847*x^4 + 373*x^3 - 6*x^2 - 17*x + 1

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:	$[14, 1]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-210271325920312500000000$ -210271325920312500000000 $\medspace = -\,2^{8}\cdot 3^{8}\cdot 5^{14}\cdot 29^{5}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$28.69$	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:	$2^{15/8}3^{1/2}5^{7/8}29^{3/4}\approx 324.6303510881772$
Ramified primes:	$2$, $3$, $5$, $29$ 2, 3, 5, 29	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-29}) $
$\card{ \Aut(K/\Q) }$:	$2$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{22}a^{14}+\frac{2}{11}a^{13}-\frac{5}{22}a^{12}-\frac{5}{22}a^{11}-\frac{1}{22}a^{10}+\frac{7}{22}a^{8}-\frac{2}{11}a^{7}-\frac{9}{22}a^{6}+\frac{5}{22}a^{5}-\frac{1}{11}a^{4}+\frac{3}{11}a^{3}-\frac{1}{22}a^{2}+\frac{5}{11}a+\frac{1}{22}$, $\frac{1}{3859326922}a^{15}+\frac{15732998}{1929663461}a^{14}-\frac{166778703}{1929663461}a^{13}-\frac{479421911}{3859326922}a^{12}-\frac{300524683}{3859326922}a^{11}-\frac{87727974}{1929663461}a^{10}-\frac{936571211}{3859326922}a^{9}-\frac{426093605}{3859326922}a^{8}-\frac{488066477}{3859326922}a^{7}-\frac{1844679167}{3859326922}a^{6}-\frac{330795669}{3859326922}a^{5}+\frac{916975687}{3859326922}a^{4}+\frac{317810203}{1929663461}a^{3}+\frac{564228622}{1929663461}a^{2}-\frac{1382581977}{3859326922}a+\frac{1154732499}{3859326922}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/2*a^12 - 1/2*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^4 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^13 - 1/2*a^8 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2, 1/22*a^14 + 2/11*a^13 - 5/22*a^12 - 5/22*a^11 - 1/22*a^10 + 7/22*a^8 - 2/11*a^7 - 9/22*a^6 + 5/22*a^5 - 1/11*a^4 + 3/11*a^3 - 1/22*a^2 + 5/11*a + 1/22, 1/3859326922*a^15 + 15732998/1929663461*a^14 - 166778703/1929663461*a^13 - 479421911/3859326922*a^12 - 300524683/3859326922*a^11 - 87727974/1929663461*a^10 - 936571211/3859326922*a^9 - 426093605/3859326922*a^8 - 488066477/3859326922*a^7 - 1844679167/3859326922*a^6 - 330795669/3859326922*a^5 + 916975687/3859326922*a^4 + 317810203/1929663461*a^3 + 564228622/1929663461*a^2 - 1382581977/3859326922*a + 1154732499/3859326922

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$ (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:	$14$	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:	$\frac{4193408797}{175423951}a^{15}-\frac{1852533111}{175423951}a^{14}-\frac{101677772844}{175423951}a^{13}+\frac{136141508594}{175423951}a^{12}+\frac{801501672414}{175423951}a^{11}-\frac{1674580912485}{175423951}a^{10}-\frac{1924713677443}{175423951}a^{9}+\frac{6697305527601}{175423951}a^{8}-\frac{1260123650382}{175423951}a^{7}-\frac{7987782550305}{175423951}a^{6}+\frac{5173816887081}{175423951}a^{5}+\frac{2149628918394}{175423951}a^{4}-\frac{2358917012863}{175423951}a^{3}+\frac{248099843274}{175423951}a^{2}+\frac{115676331189}{175423951}a-\frac{7484905978}{175423951}$, $\frac{23903892409}{3859326922}a^{15}-\frac{841085003}{350847902}a^{14}-\frac{579725277163}{3859326922}a^{13}+\frac{744073797097}{3859326922}a^{12}+\frac{4599988388025}{3859326922}a^{11}-\frac{4639980229949}{1929663461}a^{10}-\frac{5700206135868}{1929663461}a^{9}+\frac{18693105278326}{1929663461}a^{8}-\frac{5355135563111}{3859326922}a^{7}-\frac{22570963705396}{1929663461}a^{6}+\frac{2458630319543}{350847902}a^{5}+\frac{12812067941115}{3859326922}a^{4}-\frac{12457141359457}{3859326922}a^{3}+\frac{561504204279}{1929663461}a^{2}+\frac{299208403799}{1929663461}a-\frac{20236225583}{1929663461}$, $\frac{9194293910}{1929663461}a^{15}-\frac{381661045}{175423951}a^{14}-\frac{446451625061}{3859326922}a^{13}+\frac{603047577707}{3859326922}a^{12}+\frac{3522167814237}{3859326922}a^{11}-\frac{7397797606317}{3859326922}a^{10}-\frac{8468705269687}{3859326922}a^{9}+\frac{14801168544337}{1929663461}a^{8}-\frac{2750347951127}{1929663461}a^{7}-\frac{17745402619887}{1929663461}a^{6}+\frac{2065553995841}{350847902}a^{5}+\frac{4912076388568}{1929663461}a^{4}-\frac{10388696351555}{3859326922}a^{3}+\frac{968001879905}{3859326922}a^{2}+\frac{513409175555}{3859326922}a-\frac{10902274533}{1929663461}$, $\frac{68351101125}{3859326922}a^{15}-\frac{2863981219}{350847902}a^{14}-\frac{1657185725405}{3859326922}a^{13}+\frac{2251039391971}{3859326922}a^{12}+\frac{13033048405083}{3859326922}a^{11}-\frac{13780409807386}{1929663461}a^{10}-\frac{15471644316005}{1929663461}a^{9}+\frac{54977255525285}{1929663461}a^{8}-\frac{22367584745293}{3859326922}a^{7}-\frac{65294644347959}{1929663461}a^{6}+\frac{7889003454619}{350847902}a^{5}+\frac{34479768263553}{3859326922}a^{4}-\frac{39439032923529}{3859326922}a^{3}+\frac{2167594071735}{1929663461}a^{2}+\frac{973231239280}{1929663461}a-\frac{62097740175}{1929663461}$, $\frac{5515304589}{3859326922}a^{15}-\frac{77762913}{350847902}a^{14}-\frac{66636826051}{1929663461}a^{13}+\frac{70513109695}{1929663461}a^{12}+\frac{538910286894}{1929663461}a^{11}-\frac{1882162853581}{3859326922}a^{10}-\frac{2931707002049}{3859326922}a^{9}+\frac{3891936733989}{1929663461}a^{8}+\frac{145560339143}{3859326922}a^{7}-\frac{4825561085509}{1929663461}a^{6}+\frac{196538161851}{175423951}a^{5}+\frac{2987915163979}{3859326922}a^{4}-\frac{1034222503951}{1929663461}a^{3}+\frac{155006528653}{3859326922}a^{2}+\frac{85007632043}{3859326922}a-\frac{7404287589}{1929663461}$, $\frac{18423653858}{1929663461}a^{15}-\frac{16759298405}{3859326922}a^{14}-\frac{893111259779}{3859326922}a^{13}+\frac{54923086839}{175423951}a^{12}+\frac{3512418777239}{1929663461}a^{11}-\frac{7406214383255}{1929663461}a^{10}-\frac{16700079334055}{3859326922}a^{9}+\frac{5373357072337}{350847902}a^{8}-\frac{5965888571459}{1929663461}a^{7}-\frac{70155090182191}{3859326922}a^{6}+\frac{23288956581339}{1929663461}a^{5}+\frac{9227174864673}{1929663461}a^{4}-\frac{21159772576155}{3859326922}a^{3}+\frac{107116776605}{175423951}a^{2}+\frac{1041632565697}{3859326922}a-\frac{68351101125}{3859326922}$, $\frac{62993079647}{3859326922}a^{15}-\frac{2664227055}{350847902}a^{14}-\frac{1527904054531}{3859326922}a^{13}+\frac{2080004705549}{3859326922}a^{12}+\frac{12019157293695}{3859326922}a^{11}-\frac{12720929911860}{1929663461}a^{10}-\frac{14275546503103}{1929663461}a^{9}+\frac{50732633304063}{1929663461}a^{8}-\frac{20519138053987}{3859326922}a^{7}-\frac{60232136450606}{1929663461}a^{6}+\frac{7246188762167}{350847902}a^{5}+\frac{31741384554625}{3859326922}a^{4}-\frac{36040386450275}{3859326922}a^{3}+\frac{2026252634273}{1929663461}a^{2}+\frac{859110049382}{1929663461}a-\frac{58576030024}{1929663461}$, $\frac{56135426004}{1929663461}a^{15}-\frac{47545229399}{3859326922}a^{14}-\frac{2722892462865}{3859326922}a^{13}+\frac{1796984961718}{1929663461}a^{12}+\frac{10758956489692}{1929663461}a^{11}-\frac{22204379854617}{1929663461}a^{10}-\frac{52294170493791}{3859326922}a^{9}+\frac{178056323058807}{3859326922}a^{8}-\frac{15255140684925}{1929663461}a^{7}-\frac{213320052050879}{3859326922}a^{6}+\frac{67032780621732}{1929663461}a^{5}+\frac{29290385210902}{1929663461}a^{4}-\frac{61159141980379}{3859326922}a^{3}+\frac{3066296584151}{1929663461}a^{2}+\frac{2896008153807}{3859326922}a-\frac{191376320567}{3859326922}$, $\frac{2534392263}{350847902}a^{15}-\frac{5816358134}{1929663461}a^{14}-\frac{677378009153}{3859326922}a^{13}+\frac{443373835771}{1929663461}a^{12}+\frac{2687726926168}{1929663461}a^{11}-\frac{10985820653393}{3859326922}a^{10}-\frac{603703926271}{175423951}a^{9}+\frac{44209646427979}{3859326922}a^{8}-\frac{6486075822073}{3859326922}a^{7}-\frac{53553224784713}{3859326922}a^{6}+\frac{15937399032271}{1929663461}a^{5}+\frac{15479919941051}{3859326922}a^{4}-\frac{14584662054527}{3859326922}a^{3}+\frac{1213064180545}{3859326922}a^{2}+\frac{335868135475}{1929663461}a-\frac{43919750883}{3859326922}$, $\frac{43448486028}{1929663461}a^{15}-\frac{1752656029}{175423951}a^{14}-\frac{1053814665847}{1929663461}a^{13}+\frac{1412039251323}{1929663461}a^{12}+\frac{8309572840860}{1929663461}a^{11}-\frac{17360910141809}{1929663461}a^{10}-\frac{19975752638971}{1929663461}a^{9}+\frac{69425738582389}{1929663461}a^{8}-\frac{12937136808549}{1929663461}a^{7}-\frac{82803100156002}{1929663461}a^{6}+\frac{4852409540855}{175423951}a^{5}+\frac{22276726247870}{1929663461}a^{4}-\frac{24248763904866}{1929663461}a^{3}+\frac{2587756838552}{1929663461}a^{2}+\frac{1158318453181}{1929663461}a-\frac{80741919068}{1929663461}$, $\frac{24588787373}{1929663461}a^{15}-\frac{24646585059}{3859326922}a^{14}-\frac{1192578366327}{3859326922}a^{13}+\frac{1667139376603}{3859326922}a^{12}+\frac{9339882496767}{3859326922}a^{11}-\frac{20221907872289}{3859326922}a^{10}-\frac{10850388508592}{1929663461}a^{9}+\frac{40139733040078}{1929663461}a^{8}-\frac{9258526665367}{1929663461}a^{7}-\frac{94555025074337}{3859326922}a^{6}+\frac{32778812591197}{1929663461}a^{5}+\frac{24000401796891}{3859326922}a^{4}-\frac{29476618577881}{3859326922}a^{3}+\frac{3560887729823}{3859326922}a^{2}+\frac{715727187973}{1929663461}a-\frac{46980961818}{1929663461}$, $\frac{138592028323}{3859326922}a^{15}-\frac{5522836375}{350847902}a^{14}-\frac{3360948646439}{3859326922}a^{13}+\frac{4487121193361}{3859326922}a^{12}+\frac{26511160385685}{3859326922}a^{11}-\frac{27620615944128}{1929663461}a^{10}-\frac{31928280673038}{1929663461}a^{9}+\frac{110527479371117}{1929663461}a^{8}-\frac{40639877371141}{3859326922}a^{7}-\frac{131974362916920}{1929663461}a^{6}+\frac{15414144435341}{350847902}a^{5}+\frac{71401953097929}{3859326922}a^{4}-\frac{77251912562113}{3859326922}a^{3}+\frac{4029913968989}{1929663461}a^{2}+\frac{1874349137086}{1929663461}a-\frac{123707790307}{1929663461}$, $\frac{853730194}{175423951}a^{15}-\frac{8684142857}{3859326922}a^{14}-\frac{227842661595}{1929663461}a^{13}+\frac{309361589401}{1929663461}a^{12}+\frac{1793895377721}{1929663461}a^{11}-\frac{3786993810229}{1929663461}a^{10}-\frac{777856730671}{350847902}a^{9}+\frac{15113474961364}{1929663461}a^{8}-\frac{2969548539739}{1929663461}a^{7}-\frac{35950903998261}{3859326922}a^{6}+\frac{23485018596971}{3859326922}a^{5}+\frac{9554694729711}{3859326922}a^{4}-\frac{5281955139696}{1929663461}a^{3}+\frac{589809363096}{1929663461}a^{2}+\frac{473889424453}{3859326922}a-\frac{16248070066}{1929663461}$, $\frac{60464548032}{1929663461}a^{15}-\frac{50391619941}{3859326922}a^{14}-\frac{1465772981087}{1929663461}a^{13}+\frac{1926385420305}{1929663461}a^{12}+\frac{11582360564618}{1929663461}a^{11}-\frac{23837357059968}{1929663461}a^{10}-\frac{56323524367215}{3859326922}a^{9}+\frac{95591334237755}{1929663461}a^{8}-\frac{16338263554467}{1929663461}a^{7}-\frac{228829598354703}{3859326922}a^{6}+\frac{144090587404265}{3859326922}a^{5}+\frac{62542205572511}{3859326922}a^{4}-\frac{32893680312100}{1929663461}a^{3}+\frac{3334098491725}{1929663461}a^{2}+\frac{3151851882437}{3859326922}a-\frac{103294982570}{1929663461}$ 4193408797/175423951a^15 - 1852533111/175423951a^14 - 101677772844/175423951a^13 + 136141508594/175423951a^12 + 801501672414/175423951a^11 - 1674580912485/175423951a^10 - 1924713677443/175423951a^9 + 6697305527601/175423951a^8 - 1260123650382/175423951a^7 - 7987782550305/175423951a^6 + 5173816887081/175423951a^5 + 2149628918394/175423951a^4 - 2358917012863/175423951a^3 + 248099843274/175423951a^2 + 115676331189/175423951a - 7484905978/175423951, 23903892409/3859326922a^15 - 841085003/350847902a^14 - 579725277163/3859326922a^13 + 744073797097/3859326922a^12 + 4599988388025/3859326922a^11 - 4639980229949/1929663461a^10 - 5700206135868/1929663461a^9 + 18693105278326/1929663461a^8 - 5355135563111/3859326922a^7 - 22570963705396/1929663461a^6 + 2458630319543/350847902a^5 + 12812067941115/3859326922a^4 - 12457141359457/3859326922a^3 + 561504204279/1929663461a^2 + 299208403799/1929663461a - 20236225583/1929663461, 9194293910/1929663461a^15 - 381661045/175423951a^14 - 446451625061/3859326922a^13 + 603047577707/3859326922a^12 + 3522167814237/3859326922a^11 - 7397797606317/3859326922a^10 - 8468705269687/3859326922a^9 + 14801168544337/1929663461a^8 - 2750347951127/1929663461a^7 - 17745402619887/1929663461a^6 + 2065553995841/350847902a^5 + 4912076388568/1929663461a^4 - 10388696351555/3859326922a^3 + 968001879905/3859326922a^2 + 513409175555/3859326922a - 10902274533/1929663461, 68351101125/3859326922a^15 - 2863981219/350847902a^14 - 1657185725405/3859326922a^13 + 2251039391971/3859326922a^12 + 13033048405083/3859326922a^11 - 13780409807386/1929663461a^10 - 15471644316005/1929663461a^9 + 54977255525285/1929663461a^8 - 22367584745293/3859326922a^7 - 65294644347959/1929663461a^6 + 7889003454619/350847902a^5 + 34479768263553/3859326922a^4 - 39439032923529/3859326922a^3 + 2167594071735/1929663461a^2 + 973231239280/1929663461a - 62097740175/1929663461, 5515304589/3859326922a^15 - 77762913/350847902a^14 - 66636826051/1929663461a^13 + 70513109695/1929663461a^12 + 538910286894/1929663461a^11 - 1882162853581/3859326922a^10 - 2931707002049/3859326922a^9 + 3891936733989/1929663461a^8 + 145560339143/3859326922a^7 - 4825561085509/1929663461a^6 + 196538161851/175423951a^5 + 2987915163979/3859326922a^4 - 1034222503951/1929663461a^3 + 155006528653/3859326922a^2 + 85007632043/3859326922a - 7404287589/1929663461, 18423653858/1929663461a^15 - 16759298405/3859326922a^14 - 893111259779/3859326922a^13 + 54923086839/175423951a^12 + 3512418777239/1929663461a^11 - 7406214383255/1929663461a^10 - 16700079334055/3859326922a^9 + 5373357072337/350847902a^8 - 5965888571459/1929663461a^7 - 70155090182191/3859326922a^6 + 23288956581339/1929663461a^5 + 9227174864673/1929663461a^4 - 21159772576155/3859326922a^3 + 107116776605/175423951a^2 + 1041632565697/3859326922a - 68351101125/3859326922, 62993079647/3859326922a^15 - 2664227055/350847902a^14 - 1527904054531/3859326922a^13 + 2080004705549/3859326922a^12 + 12019157293695/3859326922a^11 - 12720929911860/1929663461a^10 - 14275546503103/1929663461a^9 + 50732633304063/1929663461a^8 - 20519138053987/3859326922a^7 - 60232136450606/1929663461a^6 + 7246188762167/350847902a^5 + 31741384554625/3859326922a^4 - 36040386450275/3859326922a^3 + 2026252634273/1929663461a^2 + 859110049382/1929663461a - 58576030024/1929663461, 56135426004/1929663461a^15 - 47545229399/3859326922a^14 - 2722892462865/3859326922a^13 + 1796984961718/1929663461a^12 + 10758956489692/1929663461a^11 - 22204379854617/1929663461a^10 - 52294170493791/3859326922a^9 + 178056323058807/3859326922a^8 - 15255140684925/1929663461a^7 - 213320052050879/3859326922a^6 + 67032780621732/1929663461a^5 + 29290385210902/1929663461a^4 - 61159141980379/3859326922a^3 + 3066296584151/1929663461a^2 + 2896008153807/3859326922a - 191376320567/3859326922, 2534392263/350847902a^15 - 5816358134/1929663461a^14 - 677378009153/3859326922a^13 + 443373835771/1929663461a^12 + 2687726926168/1929663461a^11 - 10985820653393/3859326922a^10 - 603703926271/175423951a^9 + 44209646427979/3859326922a^8 - 6486075822073/3859326922a^7 - 53553224784713/3859326922a^6 + 15937399032271/1929663461a^5 + 15479919941051/3859326922a^4 - 14584662054527/3859326922a^3 + 1213064180545/3859326922a^2 + 335868135475/1929663461a - 43919750883/3859326922, 43448486028/1929663461a^15 - 1752656029/175423951a^14 - 1053814665847/1929663461a^13 + 1412039251323/1929663461a^12 + 8309572840860/1929663461a^11 - 17360910141809/1929663461a^10 - 19975752638971/1929663461a^9 + 69425738582389/1929663461a^8 - 12937136808549/1929663461a^7 - 82803100156002/1929663461a^6 + 4852409540855/175423951a^5 + 22276726247870/1929663461a^4 - 24248763904866/1929663461a^3 + 2587756838552/1929663461a^2 + 1158318453181/1929663461a - 80741919068/1929663461, 24588787373/1929663461a^15 - 24646585059/3859326922a^14 - 1192578366327/3859326922a^13 + 1667139376603/3859326922a^12 + 9339882496767/3859326922a^11 - 20221907872289/3859326922a^10 - 10850388508592/1929663461a^9 + 40139733040078/1929663461a^8 - 9258526665367/1929663461a^7 - 94555025074337/3859326922a^6 + 32778812591197/1929663461a^5 + 24000401796891/3859326922a^4 - 29476618577881/3859326922a^3 + 3560887729823/3859326922a^2 + 715727187973/1929663461a - 46980961818/1929663461, 138592028323/3859326922a^15 - 5522836375/350847902a^14 - 3360948646439/3859326922a^13 + 4487121193361/3859326922a^12 + 26511160385685/3859326922a^11 - 27620615944128/1929663461a^10 - 31928280673038/1929663461a^9 + 110527479371117/1929663461a^8 - 40639877371141/3859326922a^7 - 131974362916920/1929663461a^6 + 15414144435341/350847902a^5 + 71401953097929/3859326922a^4 - 77251912562113/3859326922a^3 + 4029913968989/1929663461a^2 + 1874349137086/1929663461a - 123707790307/1929663461, 853730194/175423951a^15 - 8684142857/3859326922a^14 - 227842661595/1929663461a^13 + 309361589401/1929663461a^12 + 1793895377721/1929663461a^11 - 3786993810229/1929663461a^10 - 777856730671/350847902a^9 + 15113474961364/1929663461a^8 - 2969548539739/1929663461a^7 - 35950903998261/3859326922a^6 + 23485018596971/3859326922a^5 + 9554694729711/3859326922a^4 - 5281955139696/1929663461a^3 + 589809363096/1929663461a^2 + 473889424453/3859326922a - 16248070066/1929663461, 60464548032/1929663461a^15 - 50391619941/3859326922a^14 - 1465772981087/1929663461a^13 + 1926385420305/1929663461a^12 + 11582360564618/1929663461a^11 - 23837357059968/1929663461a^10 - 56323524367215/3859326922a^9 + 95591334237755/1929663461a^8 - 16338263554467/1929663461a^7 - 228829598354703/3859326922a^6 + 144090587404265/3859326922a^5 + 62542205572511/3859326922a^4 - 32893680312100/1929663461a^3 + 3334098491725/1929663461a^2 + 3151851882437/3859326922a - 103294982570/1929663461 (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:	$ 1622611.50407 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{1}\cdot 1622611.50407 \cdot 1}{2\cdot\sqrt{210271325920312500000000}}\cr\approx \mathstrut & 0.182135388352 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 24*x^14 + 46*x^13 + 173*x^12 - 506*x^11 - 236*x^10 + 1853*x^9 - 1192*x^8 - 1736*x^7 + 2296*x^6 - 177*x^5 - 847*x^4 + 373*x^3 - 6*x^2 - 17*x + 1)

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 - 24*x^14 + 46*x^13 + 173*x^12 - 506*x^11 - 236*x^10 + 1853*x^9 - 1192*x^8 - 1736*x^7 + 2296*x^6 - 177*x^5 - 847*x^4 + 373*x^3 - 6*x^2 - 17*x + 1, 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 - 24*x^14 + 46*x^13 + 173*x^12 - 506*x^11 - 236*x^10 + 1853*x^9 - 1192*x^8 - 1736*x^7 + 2296*x^6 - 177*x^5 - 847*x^4 + 373*x^3 - 6*x^2 - 17*x + 1);

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 - 24*x^14 + 46*x^13 + 173*x^12 - 506*x^11 - 236*x^10 + 1853*x^9 - 1192*x^8 - 1736*x^7 + 2296*x^6 - 177*x^5 - 847*x^4 + 373*x^3 - 6*x^2 - 17*x + 1);

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

$C_2^2\wr C_2^2.C_4$ (as 16T1616):

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 4096

The 94 conjugacy class representatives for $C_2^2\wr C_2^2.C_4$

Character table for $C_2^2\wr C_2^2.C_4$

Intermediate fields

$\Q(\sqrt{5}) $, 4.4.32625.1, $\Q(\zeta_{15})^+$, 4.4.725.1, 8.8.1064390625.1

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]

Sibling fields

Degree 16 siblings:	data not computed
Degree 32 siblings:	data not computed
Minimal sibling:	16.12.19272275600400000000000000.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$	${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	${\href{/padicField/13.8.0.1}{8} }^{2}$	${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$	${\href{/padicField/19.2.0.1}{2} }^{8}$	${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$	R	${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$	${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$	${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$	${\href{/padicField/43.8.0.1}{8} }^{2}$	${\href{/padicField/47.8.0.1}{8} }^{2}$	${\href{/padicField/53.8.0.1}{8} }^{2}$	${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}$

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
$2$ 2	2.8.8.8	$x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328$	$2$	$4$	$8$	$((C_8 : C_2):C_2):C_2$	$[2, 2, 2, 2]^{4}$
$2$ 2	2.8.0.1	$x^{8} + x^{4} + x^{3} + x^{2} + 1$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$
$3$ 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	5.16.14.1	$x^{16} - 20 x^{8} - 100$	$8$	$2$	$14$	$C_8: C_2$	$[\ ]_{8}^{2}$
$29$ 29	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.4.3.4	$x^{4} + 232$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	29.4.0.1	$x^{4} + 2 x^{2} + 15 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$

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