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Properties

Label 16.8.90087898081...1561.1
Degree $16$
Signature $[8, 4]$
Discriminant $53^{14}\cdot 97^{14}$
Root discriminant $1766.72$
Ramified primes $53, 97$
Class number $4$ (GRH)
Class group $[4]$ (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![-2821473341718922999, -946683758562289055, 149332680084096920, 54988959641645836, -2974204871463126, -915313131098278, 15967753823857, 6818143175898, 135910932348, -14840411598, -618113010, 13402587, 1114084, -10282, -1304, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1304*x^14 - 10282*x^13 + 1114084*x^12 + 13402587*x^11 - 618113010*x^10 - 14840411598*x^9 + 135910932348*x^8 + 6818143175898*x^7 + 15967753823857*x^6 - 915313131098278*x^5 - 2974204871463126*x^4 + 54988959641645836*x^3 + 149332680084096920*x^2 - 946683758562289055*x - 2821473341718922999)
 
gp: K = bnfinit(x^16 - 1304*x^14 - 10282*x^13 + 1114084*x^12 + 13402587*x^11 - 618113010*x^10 - 14840411598*x^9 + 135910932348*x^8 + 6818143175898*x^7 + 15967753823857*x^6 - 915313131098278*x^5 - 2974204871463126*x^4 + 54988959641645836*x^3 + 149332680084096920*x^2 - 946683758562289055*x - 2821473341718922999, 1)
 

Normalized defining polynomial

\( x^{16} - 1304 x^{14} - 10282 x^{13} + 1114084 x^{12} + 13402587 x^{11} - 618113010 x^{10} - 14840411598 x^{9} + 135910932348 x^{8} + 6818143175898 x^{7} + 15967753823857 x^{6} - 915313131098278 x^{5} - 2974204871463126 x^{4} + 54988959641645836 x^{3} + 149332680084096920 x^{2} - 946683758562289055 x - 2821473341718922999 \)

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:  \(9008789808170658548806656862365268212872206896031561=53^{14}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1766.72$
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:  $53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{47} a^{11} - \frac{10}{47} a^{10} + \frac{22}{47} a^{9} - \frac{20}{47} a^{8} + \frac{19}{47} a^{7} + \frac{8}{47} a^{6} + \frac{10}{47} a^{5} + \frac{21}{47} a^{4} - \frac{7}{47} a^{3} + \frac{21}{47} a^{2} + \frac{4}{47} a - \frac{1}{47}$, $\frac{1}{47} a^{12} + \frac{16}{47} a^{10} + \frac{12}{47} a^{9} + \frac{7}{47} a^{8} + \frac{10}{47} a^{7} - \frac{4}{47} a^{6} - \frac{20}{47} a^{5} + \frac{15}{47} a^{4} - \frac{2}{47} a^{3} - \frac{21}{47} a^{2} - \frac{8}{47} a - \frac{10}{47}$, $\frac{1}{47} a^{13} - \frac{16}{47} a^{10} - \frac{16}{47} a^{9} + \frac{1}{47} a^{8} + \frac{21}{47} a^{7} - \frac{7}{47} a^{6} - \frac{4}{47} a^{5} - \frac{9}{47} a^{4} - \frac{3}{47} a^{3} - \frac{15}{47} a^{2} + \frac{20}{47} a + \frac{16}{47}$, $\frac{1}{47} a^{14} + \frac{12}{47} a^{10} - \frac{23}{47} a^{9} - \frac{17}{47} a^{8} + \frac{15}{47} a^{7} - \frac{17}{47} a^{6} + \frac{10}{47} a^{5} + \frac{4}{47} a^{4} + \frac{14}{47} a^{3} - \frac{20}{47} a^{2} - \frac{14}{47} a - \frac{16}{47}$, $\frac{1}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{15} + \frac{716090717618687890930532882769325417499553350115703452142754016960686411404659008449960703461302149334056675550917732623050}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{14} + \frac{286600771669339292329181092794157115688829688843698916025538225030736646227872999489780481447060914561268017998361544194386}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{13} + \frac{257374319026888379549719534311373037475451552150253282967686409218203900425895108048416094549374353596485515848877294324916}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{12} + \frac{623244821416774755137747835532673826436853038135625303537978611439286846072142501489809810611025364774919382440796381799953}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{11} - \frac{11667589549335495488779048440444364200018042446850534516638262224117902505460178454552162259403092465521258442158910848623557}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{10} - \frac{16511508916636560336160657855979123098146900359766977571046137230826792534542706189907849314015945567149750428694355743337748}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{9} - \frac{30852955936353614166901221012410115036371799070668708987485145294600057566633594581332618267766904368595997427596804613713223}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{8} + \frac{7427079475970436698163822533426467088536688302288207459699016693144958894118362582288034898270459853096162343042587186490111}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{7} - \frac{4312691321401905853304136777947231623677009647946498120296634157779339059323156469096872623694258949770894402663066928263285}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{6} + \frac{12433670818459582082884479319236841737251797640276236179499759282886122488847976628959277413657009640665303111317063028476127}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{5} - \frac{35827741241298176053784476464186979272246123100596782204197768159115976529899558042468892860111081684556832242550452757191695}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{4} + \frac{7010206760473069743777123110638820752132200996386815285169103312102434912752024432446008627185055806263399207253220989567901}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{3} + \frac{4066399252779602968812243364459797700725470608185565056729725618744663429639945863071094500749975574726375477166288025088113}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a^{2} - \frac{28871763427529311089612709540810953232640533597222762590855369283412746077463093031150305002668931678740955845718786396800034}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207} a + \frac{13084069654204026042905063067704589099045509182829787536089298242630562022918193256723536366907111720882424106414688447831929}{83571663137927362470252912246260975376090959183778281567550902119262683333280539478324107244550355623479537563257247059457207}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 19601320715600000000 \) (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

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{5141}) \), \(\Q(\sqrt{53}, \sqrt{97})\), 8.8.18462292327593524004841.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$53$53.8.7.1$x^{8} - 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.1$x^{8} - 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$97$97.8.7.2$x^{8} - 2425$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.2$x^{8} - 2425$$8$$1$$7$$C_8$$[\ ]_{8}$