Properties

Label 16.16.238...625.1
Degree $16$
Signature $(16, 0)$
Discriminant $2.388\times 10^{33}$
Root discriminant \(121.93\)
Ramified primes $5,7,23,41,739$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^5:(C_2\times S_4)$ (as 16T1298)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 64*x^14 - 2*x^13 + 1659*x^12 + 86*x^11 - 22291*x^10 - 1493*x^9 + 164807*x^8 + 15490*x^7 - 654940*x^6 - 114472*x^5 + 1256832*x^4 + 480544*x^3 - 899136*x^2 - 634880*x - 101120)
 
Copy content gp:K = bnfinit(y^16 - 64*y^14 - 2*y^13 + 1659*y^12 + 86*y^11 - 22291*y^10 - 1493*y^9 + 164807*y^8 + 15490*y^7 - 654940*y^6 - 114472*y^5 + 1256832*y^4 + 480544*y^3 - 899136*y^2 - 634880*y - 101120, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 64*x^14 - 2*x^13 + 1659*x^12 + 86*x^11 - 22291*x^10 - 1493*x^9 + 164807*x^8 + 15490*x^7 - 654940*x^6 - 114472*x^5 + 1256832*x^4 + 480544*x^3 - 899136*x^2 - 634880*x - 101120);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 64*x^14 - 2*x^13 + 1659*x^12 + 86*x^11 - 22291*x^10 - 1493*x^9 + 164807*x^8 + 15490*x^7 - 654940*x^6 - 114472*x^5 + 1256832*x^4 + 480544*x^3 - 899136*x^2 - 634880*x - 101120)
 

\( x^{16} - 64 x^{14} - 2 x^{13} + 1659 x^{12} + 86 x^{11} - 22291 x^{10} - 1493 x^{9} + 164807 x^{8} + \cdots - 101120 \) 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:   \(2388010935714632452515275531640625\) \(\medspace = 5^{8}\cdot 7^{2}\cdot 23^{6}\cdot 41^{4}\cdot 739^{4}\) 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:  \(121.93\)
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:  $5^{1/2}7^{1/2}23^{3/4}41^{1/2}739^{1/2}\approx 10815.436373836514$
Ramified primes:   \(5\), \(7\), \(23\), \(41\), \(739\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{8}+\frac{3}{8}a^{7}-\frac{1}{4}a^{6}-\frac{3}{8}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{12}-\frac{1}{8}a^{9}-\frac{5}{16}a^{8}+\frac{3}{8}a^{7}-\frac{3}{16}a^{6}-\frac{5}{16}a^{5}+\frac{7}{16}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{13}-\frac{1}{16}a^{10}-\frac{5}{32}a^{9}-\frac{5}{16}a^{8}+\frac{13}{32}a^{7}+\frac{11}{32}a^{6}+\frac{7}{32}a^{5}+\frac{1}{16}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{907904}a^{14}-\frac{2039}{453952}a^{13}-\frac{1383}{113488}a^{12}-\frac{7277}{453952}a^{11}+\frac{35687}{907904}a^{10}-\frac{54293}{226976}a^{9}-\frac{199615}{907904}a^{8}-\frac{221539}{907904}a^{7}-\frac{337195}{907904}a^{6}+\frac{1898}{7093}a^{5}+\frac{31545}{113488}a^{4}-\frac{51297}{113488}a^{3}+\frac{11337}{56744}a^{2}-\frac{2335}{28372}a+\frac{5181}{14186}$, $\frac{1}{29158163152640}a^{15}+\frac{7259767}{14579081576320}a^{14}+\frac{35617214139}{3644770394080}a^{13}+\frac{96605350043}{14579081576320}a^{12}+\frac{478698292463}{29158163152640}a^{11}-\frac{22050280179}{227798149630}a^{10}-\frac{7204028446919}{29158163152640}a^{9}+\frac{1324187568601}{29158163152640}a^{8}+\frac{9394086988601}{29158163152640}a^{7}+\frac{2819934885841}{7289540788160}a^{6}-\frac{170979416217}{911192598520}a^{5}+\frac{919896283679}{3644770394080}a^{4}+\frac{177822327845}{364477039408}a^{3}+\frac{393817676817}{911192598520}a^{2}+\frac{41174848213}{91119259852}a+\frac{10037538649}{22779814963}$ 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:   $\frac{203673}{610565440}a^{15}+\frac{161807}{610565440}a^{14}-\frac{3364561}{152641360}a^{13}-\frac{168131}{7445920}a^{12}+\frac{348544909}{610565440}a^{11}+\frac{418616179}{610565440}a^{10}-\frac{4456559957}{610565440}a^{9}-\frac{2973253791}{305282720}a^{8}+\frac{451669662}{9540085}a^{7}+\frac{41939813147}{610565440}a^{6}-\frac{42407039951}{305282720}a^{5}-\frac{17568782893}{76320680}a^{4}+\frac{1456178339}{15264136}a^{3}+\frac{10962690057}{38160340}a^{2}+\frac{541336605}{3816034}a+\frac{41145783}{1908017}$, $\frac{16287913}{21678931712}a^{15}+\frac{8788423}{5419732928}a^{14}-\frac{237422051}{5419732928}a^{13}-\frac{1004349461}{10839465856}a^{12}+\frac{21963642939}{21678931712}a^{11}+\frac{22673245817}{10839465856}a^{10}-\frac{254510243383}{21678931712}a^{9}-\frac{513616345729}{21678931712}a^{8}+\frac{1516663857575}{21678931712}a^{7}+\frac{1527889955741}{10839465856}a^{6}-\frac{126543354101}{677466616}a^{5}-\frac{1125930731075}{2709866464}a^{4}+\frac{56174013213}{677466616}a^{3}+\frac{159591539753}{338733308}a^{2}+\frac{43997557005}{169366654}a+\frac{6225650381}{169366654}$, $\frac{16287913}{21678931712}a^{15}+\frac{8788423}{5419732928}a^{14}-\frac{237422051}{5419732928}a^{13}-\frac{1004349461}{10839465856}a^{12}+\frac{21963642939}{21678931712}a^{11}+\frac{22673245817}{10839465856}a^{10}-\frac{254510243383}{21678931712}a^{9}-\frac{513616345729}{21678931712}a^{8}+\frac{1516663857575}{21678931712}a^{7}+\frac{1527889955741}{10839465856}a^{6}-\frac{126543354101}{677466616}a^{5}-\frac{1125930731075}{2709866464}a^{4}+\frac{56174013213}{677466616}a^{3}+\frac{159591539753}{338733308}a^{2}+\frac{43997557005}{169366654}a+\frac{6564383689}{169366654}$, $\frac{2335493}{1354933232}a^{15}-\frac{10784953}{2709866464}a^{14}+\frac{136314365}{1354933232}a^{13}+\frac{38508461}{169366654}a^{12}-\frac{788130291}{338733308}a^{11}-\frac{13887606091}{2709866464}a^{10}+\frac{36481229191}{1354933232}a^{9}+\frac{156674887445}{2709866464}a^{8}-\frac{433683000035}{2709866464}a^{7}-\frac{925077557961}{2709866464}a^{6}+\frac{288882506097}{677466616}a^{5}+\frac{336917792391}{338733308}a^{4}-\frac{130124235405}{677466616}a^{3}-\frac{188416409663}{169366654}a^{2}-\frac{99370593405}{169366654}a-\frac{6884461539}{84683327}$, $\frac{13218795943}{14579081576320}a^{15}+\frac{2421958459}{1822385197040}a^{14}-\frac{6377509488}{113899074815}a^{13}-\frac{637126420091}{7289540788160}a^{12}+\frac{19947523498029}{14579081576320}a^{11}+\frac{16431096487617}{7289540788160}a^{10}-\frac{242794128300677}{14579081576320}a^{9}-\frac{420832004448627}{14579081576320}a^{8}+\frac{15\cdots 13}{14579081576320}a^{7}+\frac{13\cdots 51}{7289540788160}a^{6}-\frac{10\cdots 23}{3644770394080}a^{5}-\frac{562598197973029}{911192598520}a^{4}+\frac{24212292462771}{182238519704}a^{3}+\frac{337744939098981}{455596299260}a^{2}+\frac{9458292702225}{22779814963}a+\frac{1455152030934}{22779814963}$, $\frac{2164674597}{14579081576320}a^{15}-\frac{7641380137}{14579081576320}a^{14}-\frac{86770414513}{7289540788160}a^{13}+\frac{152398251671}{7289540788160}a^{12}+\frac{5162597303841}{14579081576320}a^{11}-\frac{3725961892529}{14579081576320}a^{10}-\frac{73475359663243}{14579081576320}a^{9}+\frac{2941402920311}{7289540788160}a^{8}+\frac{262899591933751}{7289540788160}a^{7}+\frac{173791809023533}{14579081576320}a^{6}-\frac{445449881122887}{3644770394080}a^{5}-\frac{34732124791273}{455596299260}a^{4}+\frac{54395588229155}{364477039408}a^{3}+\frac{123865437585313}{911192598520}a^{2}+\frac{166938841025}{91119259852}a-\frac{544261897123}{45559629926}$, $\frac{20795714949}{29158163152640}a^{15}+\frac{16044262743}{14579081576320}a^{14}-\frac{39750807421}{911192598520}a^{13}-\frac{974887666553}{14579081576320}a^{12}+\frac{30922158656027}{29158163152640}a^{11}+\frac{33768320533}{21068036960}a^{10}-\frac{377569751025531}{29158163152640}a^{9}-\frac{559485193164651}{29158163152640}a^{8}+\frac{23\cdots 09}{29158163152640}a^{7}+\frac{873303026751089}{7289540788160}a^{6}-\frac{885711214404367}{3644770394080}a^{5}-\frac{13\cdots 39}{3644770394080}a^{4}+\frac{73598320029263}{364477039408}a^{3}+\frac{405246859067543}{911192598520}a^{2}+\frac{15239764644265}{91119259852}a+\frac{418847088557}{22779814963}$, $\frac{17819558859}{29158163152640}a^{15}+\frac{5714917219}{7289540788160}a^{14}-\frac{255478587743}{7289540788160}a^{13}-\frac{693092819063}{14579081576320}a^{12}+\frac{22765736317457}{29158163152640}a^{11}+\frac{16200432991591}{14579081576320}a^{10}-\frac{246458839807341}{29158163152640}a^{9}-\frac{366507341502971}{29158163152640}a^{8}+\frac{13\cdots 69}{29158163152640}a^{7}+\frac{10\cdots 83}{14579081576320}a^{6}-\frac{387566877484117}{3644770394080}a^{5}-\frac{700457692165189}{3644770394080}a^{4}+\frac{5007917428051}{91119259852}a^{3}+\frac{44620162565607}{227798149630}a^{2}+\frac{3877450798285}{45559629926}a+\frac{322816068919}{45559629926}$, $\frac{439372912633}{29158163152640}a^{15}-\frac{188917018253}{7289540788160}a^{14}+\frac{38670806527}{42136073920}a^{13}+\frac{23504855677581}{14579081576320}a^{12}-\frac{644798224615259}{29158163152640}a^{11}-\frac{575878895023917}{14579081576320}a^{10}+\frac{77\cdots 67}{29158163152640}a^{9}+\frac{14\cdots 57}{29158163152640}a^{8}-\frac{47\cdots 43}{29158163152640}a^{7}-\frac{44\cdots 61}{14579081576320}a^{6}+\frac{509037067536222}{113899074815}a^{5}+\frac{34\cdots 13}{3644770394080}a^{4}-\frac{411709248500895}{182238519704}a^{3}-\frac{50\cdots 83}{455596299260}a^{2}-\frac{134755596847625}{22779814963}a-\frac{39276512864975}{45559629926}$, $\frac{17641907203}{29158163152640}a^{15}-\frac{24958805411}{14579081576320}a^{14}+\frac{7232576803}{227798149630}a^{13}+\frac{1247813516591}{14579081576320}a^{12}-\frac{18922006649589}{29158163152640}a^{11}-\frac{11836806065161}{7289540788160}a^{10}+\frac{191044368445957}{29158163152640}a^{9}+\frac{10367475790577}{711174711040}a^{8}-\frac{24202633053343}{711174711040}a^{7}-\frac{228622709633139}{3644770394080}a^{6}+\frac{18799757311299}{227798149630}a^{5}+\frac{404968022833803}{3644770394080}a^{4}-\frac{489711440365}{8889683888}a^{3}-\frac{30933229356011}{911192598520}a^{2}-\frac{3785303220389}{91119259852}a-\frac{1577204978349}{22779814963}$, $\frac{15806600981}{14579081576320}a^{15}+\frac{49774508229}{14579081576320}a^{14}-\frac{406005643819}{7289540788160}a^{13}-\frac{7476658089}{42136073920}a^{12}+\frac{16098665213413}{14579081576320}a^{11}+\frac{52286071977393}{14579081576320}a^{10}-\frac{894659510623}{84272147840}a^{9}-\frac{130871816279611}{3644770394080}a^{8}+\frac{182265592132589}{3644770394080}a^{7}+\frac{27\cdots 79}{14579081576320}a^{6}-\frac{322336672491681}{3644770394080}a^{5}-\frac{887110754724101}{1822385197040}a^{4}-\frac{27429958338789}{364477039408}a^{3}+\frac{431350358830359}{911192598520}a^{2}+\frac{27515344637307}{91119259852}a+\frac{2132874106085}{45559629926}$, $\frac{15779807259}{29158163152640}a^{15}+\frac{19448023293}{14579081576320}a^{14}-\frac{10681198623}{455596299260}a^{13}-\frac{958866189303}{14579081576320}a^{12}+\frac{9574265349877}{29158163152640}a^{11}+\frac{2321365331067}{1822385197040}a^{10}-\frac{28207516491861}{29158163152640}a^{9}-\frac{372669122722381}{29158163152640}a^{8}-\frac{423185938683781}{29158163152640}a^{7}+\frac{537618519196469}{7289540788160}a^{6}+\frac{439773643425113}{3644770394080}a^{5}-\frac{831175758140409}{3644770394080}a^{4}-\frac{103016039449823}{364477039408}a^{3}+\frac{200384217315793}{911192598520}a^{2}+\frac{22078281327161}{91119259852}a+\frac{1012205723841}{22779814963}$, $\frac{7149295989}{2915816315264}a^{15}-\frac{1509553071}{1457908157632}a^{14}+\frac{110798881131}{728954078816}a^{13}+\frac{98251775533}{1457908157632}a^{12}-\frac{10991293750443}{2915816315264}a^{11}-\frac{305491473519}{182238519704}a^{10}+\frac{138564528546015}{2915816315264}a^{9}+\frac{1433370905427}{71117471104}a^{8}-\frac{22674336830345}{71117471104}a^{7}-\frac{44917812346565}{364477039408}a^{6}+\frac{785104910542345}{728954078816}a^{5}+\frac{139355578612457}{364477039408}a^{4}-\frac{6647955170387}{4444841944}a^{3}-\frac{25468624495067}{45559629926}a^{2}+\frac{21975422775315}{45559629926}a+\frac{1604648370007}{22779814963}$, $\frac{17098717889}{14579081576320}a^{15}+\frac{135961274921}{14579081576320}a^{14}-\frac{267330895511}{7289540788160}a^{13}-\frac{3269108242953}{7289540788160}a^{12}+\frac{1273341892897}{14579081576320}a^{11}+\frac{119781481895317}{14579081576320}a^{10}+\frac{115917588190009}{14579081576320}a^{9}-\frac{266427793468429}{3644770394080}a^{8}-\frac{385240146062129}{3644770394080}a^{7}+\frac{49\cdots 31}{14579081576320}a^{6}+\frac{19\cdots 11}{3644770394080}a^{5}-\frac{13\cdots 19}{1822385197040}a^{4}-\frac{452301643462249}{364477039408}a^{3}+\frac{500641977806241}{911192598520}a^{2}+\frac{98718652972323}{91119259852}a+\frac{10259487836959}{45559629926}$, $\frac{416566123093}{29158163152640}a^{15}-\frac{309941148683}{7289540788160}a^{14}+\frac{5512212267251}{7289540788160}a^{13}+\frac{31546059505441}{14579081576320}a^{12}-\frac{458882781772479}{29158163152640}a^{11}-\frac{620557764320757}{14579081576320}a^{10}+\frac{47\cdots 07}{29158163152640}a^{9}+\frac{12\cdots 37}{29158163152640}a^{8}-\frac{25\cdots 43}{29158163152640}a^{7}-\frac{30\cdots 41}{14579081576320}a^{6}+\frac{39\cdots 57}{1822385197040}a^{5}+\frac{459132422271863}{88896838880}a^{4}-\frac{148414743569245}{182238519704}a^{3}-\frac{23\cdots 53}{455596299260}a^{2}-\frac{1521839665822}{555605243}a-\frac{18009076266909}{45559629926}$ 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:  \( 498121691542.39844 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 498121691542.39844 \cdot 1}{2\cdot\sqrt{2388010935714632452515275531640625}}\cr\approx \mathstrut & 0.334015971315704 \end{aligned}\] (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 - 64*x^14 - 2*x^13 + 1659*x^12 + 86*x^11 - 22291*x^10 - 1493*x^9 + 164807*x^8 + 15490*x^7 - 654940*x^6 - 114472*x^5 + 1256832*x^4 + 480544*x^3 - 899136*x^2 - 634880*x - 101120) 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 - 64*x^14 - 2*x^13 + 1659*x^12 + 86*x^11 - 22291*x^10 - 1493*x^9 + 164807*x^8 + 15490*x^7 - 654940*x^6 - 114472*x^5 + 1256832*x^4 + 480544*x^3 - 899136*x^2 - 634880*x - 101120, 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 - 64*x^14 - 2*x^13 + 1659*x^12 + 86*x^11 - 22291*x^10 - 1493*x^9 + 164807*x^8 + 15490*x^7 - 654940*x^6 - 114472*x^5 + 1256832*x^4 + 480544*x^3 - 899136*x^2 - 634880*x - 101120); 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 - 64*x^14 - 2*x^13 + 1659*x^12 + 86*x^11 - 22291*x^10 - 1493*x^9 + 164807*x^8 + 15490*x^7 - 654940*x^6 - 114472*x^5 + 1256832*x^4 + 480544*x^3 - 899136*x^2 - 634880*x - 101120); 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

\(\Q(\sqrt{5}) \), 4.4.16997.1, 8.8.180561255625.1

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 ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R R ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ R ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ R ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(7\) 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}$$$[\ ]^{12}$$
\(23\) Copy content Toggle raw display 23.4.1.0a1.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
23.4.1.0a1.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
23.2.4.6a1.2$x^{8} + 84 x^{7} + 2666 x^{6} + 38304 x^{5} + 221091 x^{4} + 191520 x^{3} + 66650 x^{2} + 10500 x + 648$$4$$2$$6$$D_4$$$[\ ]_{4}^{2}$$
\(41\) Copy content Toggle raw display 41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.4.1.0a1.1$x^{4} + 23 x + 6$$1$$4$$0$$C_4$$$[\ ]^{4}$$
41.2.2.2a1.1$x^{4} + 76 x^{3} + 1456 x^{2} + 497 x + 36$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(739\) 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)