Properties

Label 16.8.381...529.1
Degree $16$
Signature $[8, 4]$
Discriminant $3.816\times 10^{45}$
Root discriminant $706.07$
Ramified primes $23, 89$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024)
 
gp: K = bnfinit(x^16 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10858129511809024, -1453431519915008, -796637993795424, 4919298712952, 14997315506204, 995895836266, -50939514293, -14337715994, -132927939, 149101604, 1341198, -567292, 55734, 786, -521, -2, 1]);
 

\( x^{16} - 2 x^{15} - 521 x^{14} + 786 x^{13} + 55734 x^{12} - 567292 x^{11} + 1341198 x^{10} + 149101604 x^{9} - 132927939 x^{8} - 14337715994 x^{7} - 50939514293 x^{6} + 995895836266 x^{5} + 14997315506204 x^{4} + 4919298712952 x^{3} - 796637993795424 x^{2} - 1453431519915008 x + 10858129511809024 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3815787068614420330917907539305897270844547529\)\(\medspace = 23^{12}\cdot 89^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $706.07$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $23, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{512} a^{10} + \frac{1}{512} a^{9} + \frac{5}{256} a^{7} - \frac{3}{512} a^{6} + \frac{17}{512} a^{5} - \frac{3}{256} a^{4} - \frac{31}{128} a^{3} - \frac{15}{64} a^{2} + \frac{7}{16} a$, $\frac{1}{1024} a^{11} + \frac{3}{1024} a^{9} - \frac{1}{512} a^{8} + \frac{11}{1024} a^{7} - \frac{1}{256} a^{6} - \frac{51}{1024} a^{5} + \frac{31}{512} a^{4} + \frac{49}{256} a^{3} + \frac{25}{128} a^{2} - \frac{13}{32} a$, $\frac{1}{1024} a^{12} - \frac{1}{1024} a^{10} + \frac{1}{512} a^{9} + \frac{3}{1024} a^{8} + \frac{1}{256} a^{7} + \frac{9}{1024} a^{6} + \frac{1}{512} a^{5} + \frac{5}{256} a^{4} + \frac{31}{128} a^{3} + \frac{7}{32} a^{2} - \frac{1}{2} a$, $\frac{1}{90112} a^{13} + \frac{17}{90112} a^{12} + \frac{35}{90112} a^{11} + \frac{25}{90112} a^{10} + \frac{241}{90112} a^{9} + \frac{103}{90112} a^{8} - \frac{1863}{90112} a^{7} - \frac{893}{90112} a^{6} + \frac{1429}{45056} a^{5} - \frac{1341}{22528} a^{4} + \frac{2529}{11264} a^{3} + \frac{127}{2816} a^{2} - \frac{43}{88} a + \frac{3}{11}$, $\frac{1}{7929856} a^{14} - \frac{17}{3964928} a^{13} + \frac{21}{61952} a^{12} - \frac{7}{22528} a^{11} - \frac{13}{32768} a^{10} - \frac{197}{180224} a^{9} - \frac{5123}{1982464} a^{8} + \frac{25581}{991232} a^{7} + \frac{174417}{7929856} a^{6} + \frac{119799}{3964928} a^{5} - \frac{97271}{1982464} a^{4} + \frac{8457}{991232} a^{3} - \frac{4157}{247808} a^{2} - \frac{577}{7744} a - \frac{37}{121}$, $\frac{1}{64766351239824157147451175887252660479387933938923833523524612076289658713014272} a^{15} + \frac{251155063853674602225286974213142155212258212244710092592316952155193409}{64766351239824157147451175887252660479387933938923833523524612076289658713014272} a^{14} - \frac{110062030532177761847484856930706056573874286509456235289104941018472126531}{32383175619912078573725587943626330239693966969461916761762306038144829356507136} a^{13} - \frac{37233669601965764267084802555244612446267707202862676402866667600095312545}{252993559530563113857231155809580704997609116948921224701268015923006479347712} a^{12} + \frac{1428437192748337589254694934072620809229566669852255766589456977866436281841}{2943925056355643506702326176693302749063087906314719705614755094376802668773376} a^{11} - \frac{727890385895362687156821288607618638896730597561648282867630578446806538407}{2943925056355643506702326176693302749063087906314719705614755094376802668773376} a^{10} + \frac{1625521350673163225654854679914164760531218171704680148885215535039085906345}{505987119061126227714462311619161409995218233897842449402536031846012958695424} a^{9} - \frac{43400938490063826801557289041008091009430940407179846777788104386713075205143}{16191587809956039286862793971813165119846983484730958380881153019072414678253568} a^{8} - \frac{1149797367921761289054280070105921805652170916159127272019299272788866412274647}{64766351239824157147451175887252660479387933938923833523524612076289658713014272} a^{7} + \frac{1196090668063061840292525908384390574361321142352399348366236717193317364490145}{64766351239824157147451175887252660479387933938923833523524612076289658713014272} a^{6} - \frac{42872674009463300765268101787623131993852042118723969149374099290661193624507}{2943925056355643506702326176693302749063087906314719705614755094376802668773376} a^{5} - \frac{5657298544279578751712978595747199357497893622174429033369610402479141114841}{1471962528177821753351163088346651374531543953157359852807377547188401334386688} a^{4} - \frac{54412127519888098058118971831555245601942561148067710401404353179950006083691}{735981264088910876675581544173325687265771976578679926403688773594200667193344} a^{3} + \frac{287615473744015471244868763143660113961665746441474791107779866241043411584201}{2023948476244504910857849246476645639980872935591369797610144127384051834781696} a^{2} - \frac{4688295995253868672803901684273436085732207453868634791259763684877567592903}{63248389882640778464307788952395176249402279237230306175317003980751619836928} a + \frac{107164645522261549057141347514890184083324388520413843948321678764625695161}{494128045958131081752404601190587314448455306540861766994664093599622029976}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5313448120150000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 5313448120150000000000 \cdot 1}{2\sqrt{3815787068614420330917907539305897270844547529}}\approx 17159.8632152005$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.12377740988499730889.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ R $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
23Data not computed
89Data not computed