Properties

Label 16.16.148...000.1
Degree $16$
Signature $(16, 0)$
Discriminant $1.485\times 10^{33}$
Root discriminant \(118.37\)
Ramified primes $2,5,7,41,18097$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^5:(C_2\times S_4)$ (as 16T1298)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 66*x^14 + 61*x^13 + 1779*x^12 - 1490*x^11 - 25104*x^10 + 18332*x^9 + 197252*x^8 - 116352*x^7 - 844688*x^6 + 343904*x^5 + 1770384*x^4 - 360512*x^3 - 1355456*x^2 + 173568*x + 287744)
 
Copy content gp:K = bnfinit(y^16 - y^15 - 66*y^14 + 61*y^13 + 1779*y^12 - 1490*y^11 - 25104*y^10 + 18332*y^9 + 197252*y^8 - 116352*y^7 - 844688*y^6 + 343904*y^5 + 1770384*y^4 - 360512*y^3 - 1355456*y^2 + 173568*y + 287744, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 66*x^14 + 61*x^13 + 1779*x^12 - 1490*x^11 - 25104*x^10 + 18332*x^9 + 197252*x^8 - 116352*x^7 - 844688*x^6 + 343904*x^5 + 1770384*x^4 - 360512*x^3 - 1355456*x^2 + 173568*x + 287744);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 66*x^14 + 61*x^13 + 1779*x^12 - 1490*x^11 - 25104*x^10 + 18332*x^9 + 197252*x^8 - 116352*x^7 - 844688*x^6 + 343904*x^5 + 1770384*x^4 - 360512*x^3 - 1355456*x^2 + 173568*x + 287744)
 

\( x^{16} - x^{15} - 66 x^{14} + 61 x^{13} + 1779 x^{12} - 1490 x^{11} - 25104 x^{10} + 18332 x^{9} + \cdots + 287744 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(16, 0)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(1485107367098534320656820900000000\) \(\medspace = 2^{8}\cdot 5^{8}\cdot 7^{2}\cdot 41^{4}\cdot 18097^{4}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(118.37\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}5^{1/2}7^{1/2}41^{1/2}18097^{1/2}\approx 14413.658799902265$
Ramified primes:   \(2\), \(5\), \(7\), \(41\), \(18097\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{4}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{9}+\frac{1}{8}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{7}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{12}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}+\frac{1}{16}a^{7}-\frac{1}{8}a^{5}$, $\frac{1}{654848}a^{14}-\frac{7531}{654848}a^{13}+\frac{2973}{163712}a^{12}+\frac{19965}{654848}a^{11}+\frac{37393}{654848}a^{10}-\frac{2117}{163712}a^{9}-\frac{1267}{20464}a^{8}+\frac{14559}{163712}a^{7}-\frac{26005}{163712}a^{6}+\frac{20221}{81856}a^{5}+\frac{3771}{10232}a^{4}+\frac{6573}{20464}a^{3}+\frac{11189}{40928}a^{2}-\frac{9643}{20464}a-\frac{2545}{5116}$, $\frac{1}{4971281211392}a^{15}+\frac{2309179}{4971281211392}a^{14}+\frac{19908517257}{2485640605696}a^{13}+\frac{128067293173}{4971281211392}a^{12}+\frac{56625535583}{4971281211392}a^{11}+\frac{74101224441}{2485640605696}a^{10}+\frac{24042518661}{621410151424}a^{9}+\frac{50018165695}{1242820302848}a^{8}-\frac{12097758363}{1242820302848}a^{7}+\frac{28453460899}{310705075712}a^{6}+\frac{10750933019}{310705075712}a^{5}+\frac{35819238661}{155352537856}a^{4}+\frac{54002014097}{310705075712}a^{3}-\frac{4457529569}{38838134464}a^{2}+\frac{37220298029}{77676268928}a-\frac{5635013415}{19419067232}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $15$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{61}{10477568}a^{15}+\frac{1343}{10477568}a^{14}-\frac{2531}{5238784}a^{13}-\frac{74463}{10477568}a^{12}+\frac{139027}{10477568}a^{11}+\frac{802925}{5238784}a^{10}-\frac{195719}{1309696}a^{9}-\frac{4199677}{2619392}a^{8}+\frac{1292257}{2619392}a^{7}+\frac{5338267}{654848}a^{6}+\frac{1636455}{654848}a^{5}-\frac{5347311}{327424}a^{4}-\frac{10329875}{654848}a^{3}-\frac{8287}{81856}a^{2}+\frac{1387633}{163712}a+\frac{198877}{40928}$, $\frac{122543933}{2485640605696}a^{15}+\frac{204103263}{2485640605696}a^{14}-\frac{3465404627}{1242820302848}a^{13}-\frac{12513401567}{2485640605696}a^{12}+\frac{154490037427}{2485640605696}a^{11}+\frac{150613242941}{1242820302848}a^{10}-\frac{213706815239}{310705075712}a^{9}-\frac{897943877949}{621410151424}a^{8}+\frac{2408681256641}{621410151424}a^{7}+\frac{1361566843507}{155352537856}a^{6}-\frac{1534548759049}{155352537856}a^{5}-\frac{1881468659151}{77676268928}a^{4}+\frac{1043925528941}{155352537856}a^{3}+\frac{401521900901}{19419067232}a^{2}+\frac{23235275777}{38838134464}a-\frac{22516974755}{9709533616}$, $\frac{501337827}{4971281211392}a^{15}-\frac{138856063}{4971281211392}a^{14}-\frac{13670041357}{2485640605696}a^{13}+\frac{2443154431}{4971281211392}a^{12}+\frac{583386580653}{4971281211392}a^{11}+\frac{38507730915}{2485640605696}a^{10}-\frac{766515545545}{621410151424}a^{9}-\frac{612178349091}{1242820302848}a^{8}+\frac{8191511812447}{1242820302848}a^{7}+\frac{1461145823629}{310705075712}a^{6}-\frac{5132476905687}{310705075712}a^{5}-\frac{2597481034289}{155352537856}a^{4}+\frac{4644973094003}{310705075712}a^{3}+\frac{591200675035}{38838134464}a^{2}-\frac{436258947745}{77676268928}a-\frac{60835758861}{19419067232}$, $\frac{394256603}{4971281211392}a^{15}-\frac{4806187049}{4971281211392}a^{14}+\frac{14614174869}{2485640605696}a^{13}+\frac{265011425449}{4971281211392}a^{12}-\frac{835095022645}{4971281211392}a^{11}-\frac{2858110312411}{2485640605696}a^{10}+\frac{1455366034257}{621410151424}a^{9}+\frac{15175039872667}{1242820302848}a^{8}-\frac{20303759825847}{1242820302848}a^{7}-\frac{20424925081813}{310705075712}a^{6}+\frac{15803306226031}{310705075712}a^{5}+\frac{25440375502233}{155352537856}a^{4}-\frac{15475734200683}{310705075712}a^{3}-\frac{5215303168979}{38838134464}a^{2}+\frac{1494319011017}{77676268928}a+\frac{556109560469}{19419067232}$, $\frac{624179329}{4971281211392}a^{15}+\frac{2179145189}{4971281211392}a^{14}+\frac{20382756103}{2485640605696}a^{13}-\frac{118229421045}{4971281211392}a^{12}-\frac{1102601944767}{4971281211392}a^{11}+\frac{1236689099863}{2485640605696}a^{10}+\frac{1983380459691}{621410151424}a^{9}-\frac{6135214803647}{1242820302848}a^{8}-\frac{32295815546117}{1242820302848}a^{7}+\frac{6869782103061}{310705075712}a^{6}+\frac{35995045521077}{310705075712}a^{5}-\frac{3771400908101}{155352537856}a^{4}-\frac{74360377995601}{310705075712}a^{3}-\frac{2868593852667}{38838134464}a^{2}+\frac{9032344832099}{77676268928}a+\frac{989223477271}{19419067232}$, $\frac{1533361373}{4971281211392}a^{15}-\frac{5915320513}{4971281211392}a^{14}-\frac{40671018355}{2485640605696}a^{13}+\frac{310654381249}{4971281211392}a^{12}+\frac{1694098565907}{4971281211392}a^{11}-\frac{3157137178723}{2485640605696}a^{10}-\frac{2211907885623}{621410151424}a^{9}+\frac{15520388694883}{1242820302848}a^{8}+\frac{25018396874081}{1242820302848}a^{7}-\frac{18800285339373}{310705075712}a^{6}-\frac{19868511218537}{310705075712}a^{5}+\frac{20212430392561}{155352537856}a^{4}+\frac{33139498701005}{310705075712}a^{3}-\frac{3292863699147}{38838134464}a^{2}-\frac{3696234669791}{77676268928}a+\frac{308165479469}{19419067232}$, $\frac{28997309}{1242820302848}a^{15}-\frac{969437109}{1242820302848}a^{14}+\frac{294660283}{621410151424}a^{13}+\frac{52522364497}{1242820302848}a^{12}-\frac{71095445649}{1242820302848}a^{11}-\frac{555904962317}{621410151424}a^{10}+\frac{194680020835}{155352537856}a^{9}+\frac{2896082885635}{310705075712}a^{8}-\frac{3568507820139}{310705075712}a^{7}-\frac{962668443453}{19419067232}a^{6}+\frac{3588089159381}{77676268928}a^{5}+\frac{4911766288841}{38838134464}a^{4}-\frac{5472393879675}{77676268928}a^{3}-\frac{2413200591183}{19419067232}a^{2}+\frac{646637108783}{19419067232}a+\frac{171923542247}{4854766808}$, $\frac{1185118943}{2485640605696}a^{15}+\frac{759665347}{2485640605696}a^{14}+\frac{36409494029}{1242820302848}a^{13}-\frac{33212394187}{2485640605696}a^{12}-\frac{1815781261625}{2485640605696}a^{11}+\frac{231946370861}{1242820302848}a^{10}+\frac{2947581030769}{310705075712}a^{9}-\frac{224174162081}{621410151424}a^{8}-\frac{42448072481315}{621410151424}a^{7}-\frac{2025843140271}{155352537856}a^{6}+\frac{41202177069279}{155352537856}a^{5}+\frac{8498256701909}{77676268928}a^{4}-\frac{73656646273215}{155352537856}a^{3}-\frac{681080680919}{2427383404}a^{2}+\frac{7920388460241}{38838134464}a+\frac{1092902925573}{9709533616}$, $\frac{1081591895}{2485640605696}a^{15}-\frac{4409793709}{2485640605696}a^{14}+\frac{35271804321}{1242820302848}a^{13}+\frac{251849504957}{2485640605696}a^{12}-\frac{1807635877993}{2485640605696}a^{11}-\frac{2837110762031}{1242820302848}a^{10}+\frac{2856004891661}{310705075712}a^{9}+\frac{15895802292887}{621410151424}a^{8}-\frac{36111990656179}{621410151424}a^{7}-\frac{22761396701533}{155352537856}a^{6}+\frac{24487086769667}{155352537856}a^{5}+\frac{29699309363853}{77676268928}a^{4}-\frac{14377577411527}{155352537856}a^{3}-\frac{5363165704957}{19419067232}a^{2}+\frac{861166752741}{38838134464}a+\frac{491072200289}{9709533616}$, $\frac{4194021977}{4971281211392}a^{15}+\frac{6234701965}{4971281211392}a^{14}+\frac{135809876743}{2485640605696}a^{13}-\frac{313632508397}{4971281211392}a^{12}-\frac{7185467876407}{4971281211392}a^{11}+\frac{2961120644599}{2485640605696}a^{10}+\frac{12448087019835}{621410151424}a^{9}-\frac{12283844204135}{1242820302848}a^{8}-\frac{192077726082861}{1242820302848}a^{7}+\frac{7789159467201}{310705075712}a^{6}+\frac{199945969045957}{310705075712}a^{5}+\frac{16144016468707}{155352537856}a^{4}-\frac{384249717925321}{310705075712}a^{3}-\frac{21443996748477}{38838134464}a^{2}+\frac{46296488988355}{77676268928}a+\frac{5603953643687}{19419067232}$, $\frac{9215383703}{2485640605696}a^{15}+\frac{15638755925}{2485640605696}a^{14}-\frac{277563975453}{1242820302848}a^{13}-\frac{943466100445}{2485640605696}a^{12}+\frac{13250448212849}{2485640605696}a^{11}+\frac{11154783172067}{1242820302848}a^{10}-\frac{19790212347937}{310705075712}a^{9}-\frac{65236490799255}{621410151424}a^{8}+\frac{244806507104523}{621410151424}a^{7}+\frac{97359721936027}{155352537856}a^{6}-\frac{180196714842559}{155352537856}a^{5}-\frac{134911006412125}{77676268928}a^{4}+\frac{195568400807703}{155352537856}a^{3}+\frac{15470153427367}{9709533616}a^{2}-\frac{18704758968577}{38838134464}a-\frac{3982597514821}{9709533616}$, $\frac{20272054859}{4971281211392}a^{15}+\frac{94682929655}{4971281211392}a^{14}+\frac{500404153853}{2485640605696}a^{13}-\frac{4940461012935}{4971281211392}a^{12}-\frac{18471895771541}{4971281211392}a^{11}+\frac{49759507230893}{2485640605696}a^{10}+\frac{19320409895721}{621410151424}a^{9}-\frac{241885016697461}{1242820302848}a^{8}-\frac{138754678362071}{1242820302848}a^{7}+\frac{290553006596207}{310705075712}a^{6}+\frac{37676300319991}{310705075712}a^{5}-\frac{319687987686263}{155352537856}a^{4}+\frac{16757963773829}{310705075712}a^{3}+\frac{63430363347667}{38838134464}a^{2}-\frac{11172446315599}{77676268928}a-\frac{6788256674835}{19419067232}$, $\frac{24610702373}{4971281211392}a^{15}+\frac{97557950153}{4971281211392}a^{14}+\frac{612089445115}{2485640605696}a^{13}-\frac{5074096817193}{4971281211392}a^{12}-\frac{23031430927275}{4971281211392}a^{11}+\frac{50930811330347}{2485640605696}a^{10}+\frac{25196637934015}{621410151424}a^{9}-\frac{246600654233755}{1242820302848}a^{8}-\frac{202873373705833}{1242820302848}a^{7}+\frac{294624161670989}{310705075712}a^{6}+\frac{81382807270945}{310705075712}a^{5}-\frac{321002083731673}{155352537856}a^{4}-\frac{34062958132181}{310705075712}a^{3}+\frac{62700917098199}{38838134464}a^{2}-\frac{6609815520761}{77676268928}a-\frac{6534480599845}{19419067232}$, $\frac{2159764059}{621410151424}a^{15}+\frac{6394085413}{621410151424}a^{14}+\frac{7456721195}{38838134464}a^{13}-\frac{330109396255}{621410151424}a^{12}-\frac{2636464918623}{621410151424}a^{11}+\frac{818782618739}{77676268928}a^{10}+\frac{1875278357245}{38838134464}a^{9}-\frac{15466197168069}{155352537856}a^{8}-\frac{47517200953573}{155352537856}a^{7}+\frac{34386952627427}{77676268928}a^{6}+\frac{21081966937465}{19419067232}a^{5}-\frac{14129949071211}{19419067232}a^{4}-\frac{73973985925343}{38838134464}a^{3}-\frac{2745660025753}{19419067232}a^{2}+\frac{2073532814209}{2427383404}a+\frac{356726543855}{1213691702}$, $\frac{6040903721}{4971281211392}a^{15}+\frac{16970187043}{4971281211392}a^{14}-\frac{189251686359}{2485640605696}a^{13}-\frac{996289374979}{4971281211392}a^{12}+\frac{9436758789671}{4971281211392}a^{11}+\frac{11511557196217}{2485640605696}a^{10}-\frac{14810013135755}{621410151424}a^{9}-\frac{66056924165097}{1242820302848}a^{8}+\frac{194100260403965}{1242820302848}a^{7}+\frac{97181350290111}{310705075712}a^{6}-\frac{153194609605173}{310705075712}a^{5}-\frac{134046491695955}{155352537856}a^{4}+\frac{182212664979609}{310705075712}a^{3}+\frac{31854687700613}{38838134464}a^{2}-\frac{18751960269267}{77676268928}a-\frac{4300842911575}{19419067232}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 961059971242.4354 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 12 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 961059971242.4354 \cdot 1}{2\cdot\sqrt{1485107367098534320656820900000000}}\cr\approx \mathstrut & 0.817187091354057 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 66*x^14 + 61*x^13 + 1779*x^12 - 1490*x^11 - 25104*x^10 + 18332*x^9 + 197252*x^8 - 116352*x^7 - 844688*x^6 + 343904*x^5 + 1770384*x^4 - 360512*x^3 - 1355456*x^2 + 173568*x + 287744) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - x^15 - 66*x^14 + 61*x^13 + 1779*x^12 - 1490*x^11 - 25104*x^10 + 18332*x^9 + 197252*x^8 - 116352*x^7 - 844688*x^6 + 343904*x^5 + 1770384*x^4 - 360512*x^3 - 1355456*x^2 + 173568*x + 287744, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 66*x^14 + 61*x^13 + 1779*x^12 - 1490*x^11 - 25104*x^10 + 18332*x^9 + 197252*x^8 - 116352*x^7 - 844688*x^6 + 343904*x^5 + 1770384*x^4 - 360512*x^3 - 1355456*x^2 + 173568*x + 287744); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 66*x^14 + 61*x^13 + 1779*x^12 - 1490*x^11 - 25104*x^10 + 18332*x^9 + 197252*x^8 - 116352*x^7 - 844688*x^6 + 343904*x^5 + 1770384*x^4 - 360512*x^3 - 1355456*x^2 + 173568*x + 287744); 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

$C_2^5:(C_2\times S_4)$ (as 16T1298):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 1536
The 62 conjugacy class representatives for $C_2^5:(C_2\times S_4)$
Character table for $C_2^5:(C_2\times S_4)$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.18097.1, 8.8.204688380625.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R R ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ R ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.2.4a2.2$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
2.2.2.4a2.2$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
\(5\) Copy content Toggle raw display 5.8.2.8a1.2$x^{16} + 2 x^{12} + 6 x^{10} + 8 x^{9} + 5 x^{8} + 6 x^{6} + 8 x^{5} + 13 x^{4} + 24 x^{3} + 28 x^{2} + 16 x + 9$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(7\) Copy content Toggle raw display 7.2.2.2a1.1$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
7.12.1.0a1.1$x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$
\(41\) Copy content Toggle raw display 41.1.2.1a1.2$x^{2} + 246$$2$$1$$1$$C_2$$$[\ ]_{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.3.1.0a1.1$x^{3} + x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
41.3.1.0a1.1$x^{3} + x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
41.3.2.3a1.2$x^{6} + 2 x^{4} + 70 x^{3} + x^{2} + 70 x + 1266$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(18097\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)