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Properties

Label 16.8.24027704049...2081.2
Degree $16$
Signature $[8, 4]$
Discriminant $53^{14}\cdot 89^{15}$
Root discriminant $2169.17$
Ramified primes $53, 89$
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![3903857622444659, 5963757538824515, 3639176903823396, 1096700690244239, 145824602336670, -6633762668540, -4577780534713, -299097667517, 21247871733, 1561814892, 123682053, 2895067, -489358, -9287, -37, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 37*x^14 - 9287*x^13 - 489358*x^12 + 2895067*x^11 + 123682053*x^10 + 1561814892*x^9 + 21247871733*x^8 - 299097667517*x^7 - 4577780534713*x^6 - 6633762668540*x^5 + 145824602336670*x^4 + 1096700690244239*x^3 + 3639176903823396*x^2 + 5963757538824515*x + 3903857622444659)
 
gp: K = bnfinit(x^16 - 4*x^15 - 37*x^14 - 9287*x^13 - 489358*x^12 + 2895067*x^11 + 123682053*x^10 + 1561814892*x^9 + 21247871733*x^8 - 299097667517*x^7 - 4577780534713*x^6 - 6633762668540*x^5 + 145824602336670*x^4 + 1096700690244239*x^3 + 3639176903823396*x^2 + 5963757538824515*x + 3903857622444659, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 37 x^{14} - 9287 x^{13} - 489358 x^{12} + 2895067 x^{11} + 123682053 x^{10} + 1561814892 x^{9} + 21247871733 x^{8} - 299097667517 x^{7} - 4577780534713 x^{6} - 6633762668540 x^{5} + 145824602336670 x^{4} + 1096700690244239 x^{3} + 3639176903823396 x^{2} + 5963757538824515 x + 3903857622444659 \)

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:  \(240277040497518512877081049134097240815202879947452081=53^{14}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2169.17$
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:  $53, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{22} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{22} a - \frac{1}{2}$, $\frac{1}{220} a^{12} + \frac{1}{55} a^{11} + \frac{1}{10} a^{10} - \frac{1}{4} a^{9} + \frac{9}{20} a^{8} - \frac{3}{10} a^{7} + \frac{7}{20} a^{5} - \frac{1}{2} a^{4} - \frac{1}{5} a^{3} + \frac{21}{220} a^{2} + \frac{7}{220} a + \frac{1}{20}$, $\frac{1}{220} a^{13} - \frac{1}{55} a^{11} - \frac{3}{20} a^{10} + \frac{9}{20} a^{9} - \frac{1}{10} a^{8} - \frac{3}{10} a^{7} - \frac{3}{20} a^{6} + \frac{1}{10} a^{5} - \frac{1}{5} a^{4} - \frac{23}{220} a^{3} - \frac{7}{20} a^{2} - \frac{7}{220} a - \frac{1}{5}$, $\frac{1}{10446913900213055320} a^{14} + \frac{5146275857235943}{10446913900213055320} a^{13} - \frac{2099427376690993}{10446913900213055320} a^{12} - \frac{97564029257097981}{10446913900213055320} a^{11} + \frac{22464980158379014}{118714930684239265} a^{10} + \frac{5148192945027041}{23742986136847853} a^{9} - \frac{34081451723322893}{949719445473914120} a^{8} + \frac{156432395734044093}{949719445473914120} a^{7} - \frac{2117826031878417}{949719445473914120} a^{6} - \frac{293565907175279311}{949719445473914120} a^{5} - \frac{479908218151785509}{2089382780042611064} a^{4} + \frac{363126083007547949}{1044691390021305532} a^{3} - \frac{4913402029925519947}{10446913900213055320} a^{2} - \frac{58836641948769071}{1305864237526631915} a + \frac{155833382262873869}{949719445473914120}$, $\frac{1}{255062124888955925639199938914247573719227558670061455410484005012738329400361090873332284840} a^{15} - \frac{4795876381940068615274076269870551223633222773589743256231412162894091111}{127531062444477962819599969457123786859613779335030727705242002506369164700180545436666142420} a^{14} + \frac{8369943158957931330344078978333286849617264010925973676392407452035302896012160919061241}{25506212488895592563919993891424757371922755867006145541048400501273832940036109087333228484} a^{13} + \frac{250168753917937118118865645133045904512986079077552452011142782490847889821312731101310739}{127531062444477962819599969457123786859613779335030727705242002506369164700180545436666142420} a^{12} - \frac{1162481816097834744617532950203791801979953323013531378859414606090036725757844248797129439}{255062124888955925639199938914247573719227558670061455410484005012738329400361090873332284840} a^{11} - \frac{1535794992619569716027412667693002551116138412395445402495030775169344926696398244809867993}{11593732949497996619963633587011253350873979939548247973203818409669924063652776857878740220} a^{10} - \frac{3442447541680554756427377047321458172829027510420203202699080263961053479643390020204333441}{23187465898995993239927267174022506701747959879096495946407636819339848127305553715757480440} a^{9} - \frac{12115772718326292454007112904026244013692700061576482704375710881220060847943810870826859}{263493930670409014090082581522983030701681362262460181209177691128861910537563110406335005} a^{8} - \frac{3952379975611376615589496136642772615120465830226321615081779136391962177887230872777444537}{11593732949497996619963633587011253350873979939548247973203818409669924063652776857878740220} a^{7} + \frac{74224617290722322632738401007617018898548873248955464479416771832729791709681326198771240}{579686647474899830998181679350562667543698996977412398660190920483496203182638842893937011} a^{6} + \frac{21137853575297176331233864301865517864449474476125837155206923587171584040665673714768887827}{127531062444477962819599969457123786859613779335030727705242002506369164700180545436666142420} a^{5} - \frac{90888433917479649546681352613672320480993262858292984659400605260553249408398491481451930317}{255062124888955925639199938914247573719227558670061455410484005012738329400361090873332284840} a^{4} - \frac{77863763621576045763550256914817738681079491490337688322125792859991643922660566842023842487}{255062124888955925639199938914247573719227558670061455410484005012738329400361090873332284840} a^{3} + \frac{18584983680712091509811402562167824195522789880164921137700116816745751299540572188925319571}{51012424977791185127839987782849514743845511734012291082096801002547665880072218174666456968} a^{2} + \frac{55544585178362115626414925779092512938692531368964525487280461328457347046519077468446037711}{255062124888955925639199938914247573719227558670061455410484005012738329400361090873332284840} a - \frac{2275434923269121869843740244662315064301107405705332187354060270095630575064888025701739463}{23187465898995993239927267174022506701747959879096495946407636819339848127305553715757480440}$

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:  \( 171078788034000000000 \) (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

\(\Q(\sqrt{89}) \), 4.4.1980257921.1, 8.8.980359279842244243492241.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R $16$

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
$53$53.8.7.4$x^{8} + 424$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.4$x^{8} + 424$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
89Data not computed