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Properties

Label 16.16.1938516851...9601.1
Degree $16$
Signature $[16, 0]$
Discriminant $37^{14}\cdot 173^{14}$
Root discriminant $2140.26$
Ramified primes $37, 173$
Class number $54$ (GRH)
Class group $[3, 3, 6]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1273986283439, -54447078181100, 323798927354430, -277710740400021, 51593522344722, 18413561633240, -5858782940122, -227735489612, 134048526390, 2176523968, -935790958, -6535626, 2419205, 7614, -2612, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 2612*x^14 + 7614*x^13 + 2419205*x^12 - 6535626*x^11 - 935790958*x^10 + 2176523968*x^9 + 134048526390*x^8 - 227735489612*x^7 - 5858782940122*x^6 + 18413561633240*x^5 + 51593522344722*x^4 - 277710740400021*x^3 + 323798927354430*x^2 - 54447078181100*x - 1273986283439)
 
gp: K = bnfinit(x^16 - 3*x^15 - 2612*x^14 + 7614*x^13 + 2419205*x^12 - 6535626*x^11 - 935790958*x^10 + 2176523968*x^9 + 134048526390*x^8 - 227735489612*x^7 - 5858782940122*x^6 + 18413561633240*x^5 + 51593522344722*x^4 - 277710740400021*x^3 + 323798927354430*x^2 - 54447078181100*x - 1273986283439, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 2612 x^{14} + 7614 x^{13} + 2419205 x^{12} - 6535626 x^{11} - 935790958 x^{10} + 2176523968 x^{9} + 134048526390 x^{8} - 227735489612 x^{7} - 5858782940122 x^{6} + 18413561633240 x^{5} + 51593522344722 x^{4} - 277710740400021 x^{3} + 323798927354430 x^{2} - 54447078181100 x - 1273986283439 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(193851685179255766530952003996759265201819225743449601=37^{14}\cdot 173^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2140.26$
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:  $37, 173$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} - \frac{8}{41} a^{11} + \frac{17}{41} a^{10} - \frac{3}{41} a^{9} + \frac{4}{41} a^{7} - \frac{3}{41} a^{6} - \frac{7}{41} a^{5} - \frac{13}{41} a^{4} - \frac{12}{41} a^{3} - \frac{11}{41} a^{2} - \frac{4}{41} a - \frac{3}{41}$, $\frac{1}{41} a^{13} - \frac{6}{41} a^{11} + \frac{10}{41} a^{10} + \frac{17}{41} a^{9} + \frac{4}{41} a^{8} - \frac{12}{41} a^{7} + \frac{10}{41} a^{6} + \frac{13}{41} a^{5} + \frac{7}{41} a^{4} + \frac{16}{41} a^{3} - \frac{10}{41} a^{2} + \frac{6}{41} a + \frac{17}{41}$, $\frac{1}{41} a^{14} + \frac{3}{41} a^{11} - \frac{4}{41} a^{10} - \frac{14}{41} a^{9} - \frac{12}{41} a^{8} - \frac{7}{41} a^{7} - \frac{5}{41} a^{6} + \frac{6}{41} a^{5} + \frac{20}{41} a^{4} - \frac{19}{41} a^{2} - \frac{7}{41} a - \frac{18}{41}$, $\frac{1}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{15} - \frac{20522656863293387712150341809604044444699407323404762561424912696021499818818235319656787164145948613099}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{14} + \frac{88211989779785773483240099319176559696078240763941959417602228804680392158496960054034207727073649473820}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{13} - \frac{96157520796397839255971920590619662374155837506218451799230723872090676448994782465189309102321909009959}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{12} - \frac{467590791928301301007004517573665141391130634773438997983528894526203705495120690692176291140652426083823}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{11} + \frac{4140348652102173662144246276998529738235114541798164549106731337491728564352671055545037401517744934296995}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{10} + \frac{969391529689985474352425500719504963031436835417554571317145502205596884412218691511992602449378573396999}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{9} + \frac{24048508390412615510335172647775316519677640451549093263856316910470060693338898937849288682806936244432}{186531484615780142295760935922292834710857407792054359499105598189366461809338432235264957700267856183317} a^{8} - \frac{663407026595397342872893325689377239684846699228920703612385137266608382443048283690633184779455770202053}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{7} - \frac{1758761004188786448324012294939347612279503889574946840492578830046921652990892554355190873944033508004508}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{6} - \frac{192770165551767322842244799658119939336315519520185214515325207003081377553317832876329050785663387548197}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{5} - \frac{3903917116910769464890852511660938863612061730092973127242019419812387064299428758079708593667233564339494}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{4} + \frac{4150951644701564160404186053456089128118761377139781568894347579485236305872643247349705535544131647397507}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{3} + \frac{3261857919911167894701964866537714770589976737123041800258344544978873501719790843410712282387360996109281}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a^{2} + \frac{3790564516430708975907049181612746817387115071303945170759331704835200719769476696374224048591913399955248}{8766979776941666687900763988347763231410298166226554896457963114900223705038906315057453011912589240615899} a + \frac{42621487026335327829553632051835848514424997696072736476527530123257869869488739873815539248064022598673}{186531484615780142295760935922292834710857407792054359499105598189366461809338432235264957700267856183317}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{6}$, which has order $54$ (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:  $15$
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:  \( 86450372936600000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{173}) \), \(\Q(\sqrt{6401}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{37}, \sqrt{173})\), 8.8.68783926416507494438401.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$173$173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$