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Properties

Label 16.0.16225116300...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 149^{14}$
Root discriminant $325.94$
Ramified primes $5, 149$
Class number $132496$ (GRH)
Class group $[2, 182, 364]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94392688351355, -16618897630070, 6088967690620, -763530138840, 220917959275, -25233283590, 6595225475, -693240910, 164477936, -14048154, 2756157, -184482, 34595, -1628, 277, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 277*x^14 - 1628*x^13 + 34595*x^12 - 184482*x^11 + 2756157*x^10 - 14048154*x^9 + 164477936*x^8 - 693240910*x^7 + 6595225475*x^6 - 25233283590*x^5 + 220917959275*x^4 - 763530138840*x^3 + 6088967690620*x^2 - 16618897630070*x + 94392688351355)
 
gp: K = bnfinit(x^16 - 6*x^15 + 277*x^14 - 1628*x^13 + 34595*x^12 - 184482*x^11 + 2756157*x^10 - 14048154*x^9 + 164477936*x^8 - 693240910*x^7 + 6595225475*x^6 - 25233283590*x^5 + 220917959275*x^4 - 763530138840*x^3 + 6088967690620*x^2 - 16618897630070*x + 94392688351355, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 277 x^{14} - 1628 x^{13} + 34595 x^{12} - 184482 x^{11} + 2756157 x^{10} - 14048154 x^{9} + 164477936 x^{8} - 693240910 x^{7} + 6595225475 x^{6} - 25233283590 x^{5} + 220917959275 x^{4} - 763530138840 x^{3} + 6088967690620 x^{2} - 16618897630070 x + 94392688351355 \)

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:  \(16225116300501260546649351778570556640625=5^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $325.94$
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:  $5, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{24} a^{12} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{24} a^{6} + \frac{5}{12} a^{5} - \frac{1}{24} a^{4} - \frac{1}{12} a^{3} + \frac{11}{24} a^{2} - \frac{1}{6} a - \frac{11}{24}$, $\frac{1}{24} a^{13} - \frac{1}{12} a^{11} + \frac{1}{6} a^{10} - \frac{1}{4} a^{9} - \frac{1}{6} a^{8} + \frac{5}{24} a^{7} - \frac{1}{6} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{12} a^{2} + \frac{3}{8} a + \frac{5}{12}$, $\frac{1}{9456} a^{14} - \frac{43}{3152} a^{13} + \frac{47}{3152} a^{12} - \frac{29}{1182} a^{11} + \frac{161}{1182} a^{10} + \frac{313}{1576} a^{9} + \frac{1051}{9456} a^{8} + \frac{955}{9456} a^{7} + \frac{463}{4728} a^{6} + \frac{1271}{9456} a^{5} + \frac{223}{591} a^{4} - \frac{97}{3152} a^{3} - \frac{361}{1182} a^{2} - \frac{1711}{9456} a - \frac{3691}{9456}$, $\frac{1}{39099062451058640165206387768206606842407433671333746032033979625761259703232622468656} a^{15} + \frac{568544191647791577547598570096642649445080434119830443083172977234223452661544313}{19549531225529320082603193884103303421203716835666873016016989812880629851616311234328} a^{14} + \frac{186279904309155970436725344779806806222034067862354396848690118880055882051295994571}{9774765612764660041301596942051651710601858417833436508008494906440314925808155617164} a^{13} - \frac{181690868218085397876038157339482591161343292146765559450072987409842098405945111159}{39099062451058640165206387768206606842407433671333746032033979625761259703232622468656} a^{12} + \frac{460940908996754821706206922368917983241151002802862187651742808112449443957941439061}{2443691403191165010325399235512912927650464604458359127002123726610078731452038904291} a^{11} - \frac{2938613346023079607925761205043278276912737132143670346148475549846734587017418480399}{19549531225529320082603193884103303421203716835666873016016989812880629851616311234328} a^{10} - \frac{5135761719426310293884060668459274615732941926087344152895291377459875329180101172599}{39099062451058640165206387768206606842407433671333746032033979625761259703232622468656} a^{9} - \frac{181082775486744511120073983655800339648377471485905391472739593392813512844600913039}{814563801063721670108466411837637642550154868152786375667374575536692910484012968097} a^{8} + \frac{7972431051943659738387414324519820324038551030251058710369030044178661413088130719449}{39099062451058640165206387768206606842407433671333746032033979625761259703232622468656} a^{7} + \frac{1945137671816638800185817994811985113070716106493199241716953276016264847490857842785}{13033020817019546721735462589402202280802477890444582010677993208587086567744207489552} a^{6} - \frac{4157703160426116060182839329289977465685387480029571147655135594585600284522110220961}{39099062451058640165206387768206606842407433671333746032033979625761259703232622468656} a^{5} + \frac{2268609995998480787252687143283441847917294735644938482631990454542870850502864727745}{13033020817019546721735462589402202280802477890444582010677993208587086567744207489552} a^{4} + \frac{3065791201874925674010168388953525443194259980737218313470659857061678780437664952387}{13033020817019546721735462589402202280802477890444582010677993208587086567744207489552} a^{3} - \frac{15323410132989954890889478561765774491264051163394078220985551599679885318769690370837}{39099062451058640165206387768206606842407433671333746032033979625761259703232622468656} a^{2} - \frac{615782312234747936283809991593767729426457877110233213369236043126380479352977584483}{19549531225529320082603193884103303421203716835666873016016989812880629851616311234328} a - \frac{2519917590172099844959881464985039391508121482177684015874226932935280382420228854835}{39099062451058640165206387768206606842407433671333746032033979625761259703232622468656}$

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

Class group and class number

$C_{2}\times C_{182}\times C_{364}$, which has order $132496$ (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:  \( 238792977.62 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{745}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{149}) \), 4.4.413493625.1, 4.4.413493625.2, \(\Q(\sqrt{5}, \sqrt{149})\), 8.8.170976977915640625.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. 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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$149$149.8.7.1$x^{8} - 149$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.1$x^{8} - 149$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$