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Properties

Label 16.0.54602537259...4609.1
Degree $16$
Signature $[0, 8]$
Discriminant $61^{14}\cdot 157^{14}$
Root discriminant $3044.92$
Ramified primes $61, 157$
Class number $56814834050$ (GRH)
Class group $[168545, 337090]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![944525217449845297152, -2796479624717236224, -12244752595015384320, -636825223635723648, 79009337564951184, 4494586179489760, 1180694075592, -19670422590640, -413101870103, -3559324740, 1115337704, -7157146, 3184374, 35728, 3581, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 3581*x^14 + 35728*x^13 + 3184374*x^12 - 7157146*x^11 + 1115337704*x^10 - 3559324740*x^9 - 413101870103*x^8 - 19670422590640*x^7 + 1180694075592*x^6 + 4494586179489760*x^5 + 79009337564951184*x^4 - 636825223635723648*x^3 - 12244752595015384320*x^2 - 2796479624717236224*x + 944525217449845297152)
 
gp: K = bnfinit(x^16 - 6*x^15 + 3581*x^14 + 35728*x^13 + 3184374*x^12 - 7157146*x^11 + 1115337704*x^10 - 3559324740*x^9 - 413101870103*x^8 - 19670422590640*x^7 + 1180694075592*x^6 + 4494586179489760*x^5 + 79009337564951184*x^4 - 636825223635723648*x^3 - 12244752595015384320*x^2 - 2796479624717236224*x + 944525217449845297152, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 3581 x^{14} + 35728 x^{13} + 3184374 x^{12} - 7157146 x^{11} + 1115337704 x^{10} - 3559324740 x^{9} - 413101870103 x^{8} - 19670422590640 x^{7} + 1180694075592 x^{6} + 4494586179489760 x^{5} + 79009337564951184 x^{4} - 636825223635723648 x^{3} - 12244752595015384320 x^{2} - 2796479624717236224 x + 944525217449845297152 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54602537259435919051622734069846680881723505970000234609=61^{14}\cdot 157^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3044.92$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $61, 157$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{72} a^{6} + \frac{1}{24} a^{5} - \frac{1}{9} a^{4} - \frac{1}{12} a^{3} - \frac{29}{72} a^{2} - \frac{1}{12} a$, $\frac{1}{72} a^{7} - \frac{5}{72} a^{5} + \frac{1}{12} a^{4} - \frac{11}{72} a^{3} - \frac{5}{24} a^{2} + \frac{1}{12} a$, $\frac{1}{216} a^{8} + \frac{1}{216} a^{7} + \frac{1}{216} a^{6} - \frac{5}{216} a^{5} + \frac{19}{216} a^{4} - \frac{7}{108} a^{3} + \frac{11}{72} a^{2} + \frac{1}{12} a$, $\frac{1}{648} a^{9} - \frac{1}{216} a^{7} - \frac{17}{216} a^{5} + \frac{1}{8} a^{4} + \frac{11}{162} a^{3} + \frac{7}{18} a$, $\frac{1}{648} a^{10} + \frac{1}{216} a^{7} - \frac{1}{216} a^{6} - \frac{5}{216} a^{5} - \frac{43}{648} a^{4} - \frac{4}{27} a^{3} + \frac{7}{36} a^{2} - \frac{1}{3} a$, $\frac{1}{3888} a^{11} - \frac{1}{1944} a^{10} - \frac{1}{1296} a^{9} + \frac{1}{648} a^{7} + \frac{1}{162} a^{6} - \frac{161}{1944} a^{5} + \frac{53}{972} a^{4} - \frac{101}{1296} a^{3} + \frac{95}{216} a^{2} + \frac{2}{9} a$, $\frac{1}{15552} a^{12} - \frac{1}{7776} a^{11} + \frac{1}{1728} a^{10} - \frac{1}{1296} a^{9} - \frac{5}{2592} a^{8} + \frac{1}{2592} a^{7} - \frac{1}{972} a^{6} - \frac{253}{3888} a^{5} - \frac{143}{1728} a^{4} + \frac{121}{1296} a^{3} + \frac{1}{144} a^{2} - \frac{1}{36} a$, $\frac{1}{186624} a^{13} - \frac{1}{46656} a^{12} + \frac{13}{186624} a^{11} - \frac{1}{3456} a^{10} - \frac{5}{31104} a^{9} - \frac{49}{31104} a^{8} - \frac{205}{46656} a^{7} + \frac{169}{46656} a^{6} + \frac{9521}{186624} a^{5} + \frac{169}{1152} a^{4} + \frac{481}{5184} a^{3} + \frac{17}{96} a^{2} - \frac{11}{24} a - \frac{1}{2}$, $\frac{1}{1492992} a^{14} - \frac{1}{746496} a^{13} - \frac{43}{1492992} a^{12} - \frac{31}{373248} a^{11} + \frac{17}{248832} a^{10} - \frac{155}{248832} a^{9} + \frac{55}{46656} a^{8} - \frac{2545}{373248} a^{7} + \frac{2713}{1492992} a^{6} - \frac{27539}{373248} a^{5} + \frac{8405}{62208} a^{4} - \frac{137}{1728} a^{3} + \frac{719}{3456} a^{2} + \frac{125}{288} a - \frac{3}{8}$, $\frac{1}{35009521441807852915094034422501121765337580046709597819918616159368959084570843904471069085108367343791610650624} a^{15} - \frac{2206715972884083408095854695488135755157039643461884406602171409551038302338441101050901249105883009777031}{17504760720903926457547017211250560882668790023354798909959308079684479542285421952235534542554183671895805325312} a^{14} - \frac{29286569582361946822306681703867497826557676884849830006192415947273588236352481166864740193259213233354785}{11669840480602617638364678140833707255112526682236532606639538719789653028190281301490356361702789114597203550208} a^{13} - \frac{117348832817522474389857218707360693434935293273311414885616874618898815612847764852160967759888522347285295}{4376190180225981614386754302812640220667197505838699727489827019921119885571355488058883635638545917973951331328} a^{12} + \frac{1900765899480541325071926400094000920793040638564927515299559967727846832283351124146057262530687788934050003}{17504760720903926457547017211250560882668790023354798909959308079684479542285421952235534542554183671895805325312} a^{11} - \frac{256612093234451693372909495639658080752760504976255331458449992564301288889230823133420010842776840081802129}{833560034328758402740334152916693375365180477302609471902824194270689502013591521535025454407342079614085967872} a^{10} - \frac{1833464207092136233595558739109913442223444486344012219168524429210049219861234860480111738795776064830756517}{4376190180225981614386754302812640220667197505838699727489827019921119885571355488058883635638545917973951331328} a^{9} + \frac{15546538219813810554410312792468251172408984068220581087321427298916554533763852204591593895418079437249530055}{8752380360451963228773508605625280441334395011677399454979654039842239771142710976117767271277091835947902662656} a^{8} - \frac{7568125073070999542759784591145290326465117426505646080735395267617173878330399676536132480661417336860810587}{1667120068657516805480668305833386750730360954605218943805648388541379004027183043070050908814684159228171935744} a^{7} + \frac{26130274339173800783729923544114930222853466614523424358959360359027075723384022521885184945733041434171572361}{4376190180225981614386754302812640220667197505838699727489827019921119885571355488058883635638545917973951331328} a^{6} - \frac{41395596319250201429298382423304527543237779991974597055728487106733291296877495437392659031588292913903956377}{4376190180225981614386754302812640220667197505838699727489827019921119885571355488058883635638545917973951331328} a^{5} + \frac{726206142407525462859945319417779197739839503161927862394236985746954166933006754414714063360338813718315479}{17365834048515800057090294852431111986774593277137697331308837380639364625283156698646363633486293325293457664} a^{4} - \frac{4165398645680441320362295454130025859366778925759257918184093734130822137058643745028109695068862368421000463}{243121676679221200799264127934035567814844305879927762638323723328951104753964193781049090868808106554108407296} a^{3} - \frac{17892230066553984606470523858650089544221775314779292120199438185284057145179175479863492994798582880335349}{562781658979678705553852147995452703275102559907240191292416026224423853597139337456132154788907654060436128} a^{2} - \frac{4363650028725359920721632043889877765692410678120760374377315200473246581406018407296500522670769559958599}{60298034890679861309341301570941361065189559990061449067044574238331127171122071870299873727382962935046728} a + \frac{2416922066940366484592785352255718754498241045651289233805317331173935293167863384962606611717684934450409}{15632823860546630709829226333207019535419515552978894202567111839567329266587203818225893188580768168345448}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{168545}\times C_{337090}$, which has order $56814834050$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10081352048400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{9577}) \), \(\Q(\sqrt{157}) \), \(\Q(\sqrt{61}, \sqrt{157})\), 4.4.878392183033.1, 4.4.878392183033.2, 8.8.771572827213479373079089.1, 8.0.7389352966223491955978435353.1 x2

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

Sibling fields

Degree 8 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
61Data not computed
157Data not computed