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Properties

Label 16.8.87995526059...5937.2
Degree $16$
Signature $[8, 4]$
Discriminant $61^{14}\cdot 73^{15}$
Root discriminant $2037.18$
Ramified primes $61, 73$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![97623421972208003, 79631144220170561, 29656924089445914, 4255857054532135, 192307300221262, -23859158339510, -7222778406903, -374714718069, 21202851621, 1646275166, 104450131, 2577855, -466626, -8787, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 - 8787*x^13 - 466626*x^12 + 2577855*x^11 + 104450131*x^10 + 1646275166*x^9 + 21202851621*x^8 - 374714718069*x^7 - 7222778406903*x^6 - 23859158339510*x^5 + 192307300221262*x^4 + 4255857054532135*x^3 + 29656924089445914*x^2 + 79631144220170561*x + 97623421972208003)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 - 8787*x^13 - 466626*x^12 + 2577855*x^11 + 104450131*x^10 + 1646275166*x^9 + 21202851621*x^8 - 374714718069*x^7 - 7222778406903*x^6 - 23859158339510*x^5 + 192307300221262*x^4 + 4255857054532135*x^3 + 29656924089445914*x^2 + 79631144220170561*x + 97623421972208003, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} - 8787 x^{13} - 466626 x^{12} + 2577855 x^{11} + 104450131 x^{10} + 1646275166 x^{9} + 21202851621 x^{8} - 374714718069 x^{7} - 7222778406903 x^{6} - 23859158339510 x^{5} + 192307300221262 x^{4} + 4255857054532135 x^{3} + 29656924089445914 x^{2} + 79631144220170561 x + 97623421972208003 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(87995526059768479187762363921103601182604493355285937=61^{14}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2037.18$
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:  $61, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{48805782501095973464} a^{14} - \frac{5931664191177995427}{48805782501095973464} a^{13} - \frac{112512158740759511}{48805782501095973464} a^{12} - \frac{1889560776674104541}{48805782501095973464} a^{11} + \frac{4890862397023833941}{24402891250547986732} a^{10} + \frac{3447152357223395271}{12201445625273993366} a^{9} - \frac{18714637639306841407}{48805782501095973464} a^{8} - \frac{1623396512451191493}{48805782501095973464} a^{7} + \frac{16836664617814451249}{48805782501095973464} a^{6} - \frac{22415138087177949929}{48805782501095973464} a^{5} - \frac{11356078538756820703}{48805782501095973464} a^{4} - \frac{128068193320733715}{595192469525560652} a^{3} + \frac{16123868578562351425}{48805782501095973464} a^{2} + \frac{1518528023460734997}{24402891250547986732} a + \frac{10119222796559031133}{48805782501095973464}$, $\frac{1}{1017437683782216509295199163240217892817033912035336081677706769008022256327467231606536834526525193944} a^{15} - \frac{3371384941293452398828982102753103091864232814718933972993994291818007435049543495}{508718841891108254647599581620108946408516956017668040838853384504011128163733615803268417263262596972} a^{14} + \frac{30316199538052026828968176644045962150558865436634732360792984333324220620486790552996548678132470387}{508718841891108254647599581620108946408516956017668040838853384504011128163733615803268417263262596972} a^{13} + \frac{7975271919062426423644641498405751451565047799141268020806377565136046574290046948971639760279605355}{254359420945554127323799790810054473204258478008834020419426692252005564081866807901634208631631298486} a^{12} + \frac{121032491884658383267658599543725972407328085693780753679412545158919449092281076393692452225513044965}{1017437683782216509295199163240217892817033912035336081677706769008022256327467231606536834526525193944} a^{11} + \frac{24228969042049896638098644985193326860468303367893917034044198337496554554808157081060020818641597893}{508718841891108254647599581620108946408516956017668040838853384504011128163733615803268417263262596972} a^{10} + \frac{225571393615782418677505598234630408291366058854572643863070724411232471567917931021640290699367652945}{1017437683782216509295199163240217892817033912035336081677706769008022256327467231606536834526525193944} a^{9} - \frac{86228381318803767557722773852689298385765324877229836822123148388242579374562262426992459546157910687}{254359420945554127323799790810054473204258478008834020419426692252005564081866807901634208631631298486} a^{8} + \frac{28894193972350041162292472325999769028513471714351632410530948400580263639034682148496650984717862185}{254359420945554127323799790810054473204258478008834020419426692252005564081866807901634208631631298486} a^{7} - \frac{12180422821930589776165524207667458119272118256925899671308119148823348573963819618494197738409726352}{127179710472777063661899895405027236602129239004417010209713346126002782040933403950817104315815649243} a^{6} + \frac{120916869559345597646853564513193563583165978818879383242699161456590116377547970629285500905208313099}{254359420945554127323799790810054473204258478008834020419426692252005564081866807901634208631631298486} a^{5} - \frac{270928376471069586761469717260498681434278401409171870322526524155109914379219397881699877508949002545}{1017437683782216509295199163240217892817033912035336081677706769008022256327467231606536834526525193944} a^{4} + \frac{433811106776950575100538935914403442645376174488134459137825547875996305290411381441886327071757717431}{1017437683782216509295199163240217892817033912035336081677706769008022256327467231606536834526525193944} a^{3} + \frac{64144554287236012274231204285603095221687136040416920318083564416063176556390962039305066421160184551}{1017437683782216509295199163240217892817033912035336081677706769008022256327467231606536834526525193944} a^{2} + \frac{99480928886357658910854534821868343998230026932667547620786838382042176263603653354733045599870244115}{1017437683782216509295199163240217892817033912035336081677706769008022256327467231606536834526525193944} a - \frac{37347466308064407957649068021412488464268691568805647805088073659393310532404356233132775049887297191}{1017437683782216509295199163240217892817033912035336081677706769008022256327467231606536834526525193944}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $11$
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:  \( 37285833957300000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.1447532257.1, 8.8.569166107419034447672017.1

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

Sibling fields

Degree 32 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ $16$ $16$ $16$

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
$61$61.8.7.3$x^{8} + 122$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.3$x^{8} + 122$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
73Data not computed