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Properties

Label 16.8.11183757375...5601.1
Degree $16$
Signature $[8, 4]$
Discriminant $29^{10}\cdot 149^{14}$
Root discriminant $653.94$
Ramified primes $29, 149$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25067773199375, 47996844987430, -6076638595078, -6363865443214, 1610473646654, 24714866448, -30081227003, 1969112717, -321324127, 56558428, 1797989, -1061325, 44342, 4064, -423, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 423*x^14 + 4064*x^13 + 44342*x^12 - 1061325*x^11 + 1797989*x^10 + 56558428*x^9 - 321324127*x^8 + 1969112717*x^7 - 30081227003*x^6 + 24714866448*x^5 + 1610473646654*x^4 - 6363865443214*x^3 - 6076638595078*x^2 + 47996844987430*x - 25067773199375)
 
gp: K = bnfinit(x^16 - 3*x^15 - 423*x^14 + 4064*x^13 + 44342*x^12 - 1061325*x^11 + 1797989*x^10 + 56558428*x^9 - 321324127*x^8 + 1969112717*x^7 - 30081227003*x^6 + 24714866448*x^5 + 1610473646654*x^4 - 6363865443214*x^3 - 6076638595078*x^2 + 47996844987430*x - 25067773199375, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 423 x^{14} + 4064 x^{13} + 44342 x^{12} - 1061325 x^{11} + 1797989 x^{10} + 56558428 x^{9} - 321324127 x^{8} + 1969112717 x^{7} - 30081227003 x^{6} + 24714866448 x^{5} + 1610473646654 x^{4} - 6363865443214 x^{3} - 6076638595078 x^{2} + 47996844987430 x - 25067773199375 \)

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:  \(1118375737550090727607450849340740343042175601=29^{10}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $653.94$
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:  $29, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{20} a^{13} - \frac{1}{4} a^{11} - \frac{1}{20} a^{10} - \frac{3}{10} a^{9} - \frac{3}{10} a^{8} - \frac{7}{20} a^{7} - \frac{3}{10} a^{6} - \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{3}{20} a^{3} - \frac{3}{20} a^{2} - \frac{9}{20} a - \frac{1}{4}$, $\frac{1}{80} a^{14} - \frac{1}{5} a^{11} + \frac{7}{40} a^{10} - \frac{11}{80} a^{9} + \frac{19}{40} a^{8} + \frac{9}{80} a^{7} + \frac{7}{40} a^{6} - \frac{17}{40} a^{5} + \frac{17}{80} a^{4} + \frac{37}{80} a^{3} + \frac{9}{20} a^{2} - \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{111044608537878962909276479353955899162124506460377077704514742958188557024854849448952095990160} a^{15} + \frac{177037615104925066513933738539434349963051550729394441305766989992658282806820914523981729763}{55522304268939481454638239676977949581062253230188538852257371479094278512427424724476047995080} a^{14} - \frac{7627891278511741989548988514795719753834211833324807627148663609947344757690152275566621056}{6940288033617435181829779959622243697632781653773567356532171434886784814053428090559505999385} a^{13} + \frac{62355347934942331517813917433177766167148344304901873906440149830550336674006755930116954091}{4626858689078290121219853306414829131755187769182378237688114289924523209368952060373003999590} a^{12} - \frac{13715925659356558668904865096396463165714614631686732707230103144918957243158879222920901911321}{55522304268939481454638239676977949581062253230188538852257371479094278512427424724476047995080} a^{11} + \frac{12779887789493970658706433242277643654441486480211242301309428885277339951760077703040799343569}{111044608537878962909276479353955899162124506460377077704514742958188557024854849448952095990160} a^{10} + \frac{11145121358767730096993099908324717172111958462820722082088365984919843209008618954243901095217}{27761152134469740727319119838488974790531126615094269426128685739547139256213712362238023997540} a^{9} - \frac{13557789810805352671979778154607665455769752367395146304492263172907322746864292979155310138547}{111044608537878962909276479353955899162124506460377077704514742958188557024854849448952095990160} a^{8} - \frac{605614784541078524841244637839968563635371808165297259976615266595624691151817784020707985301}{5552230426893948145463823967697794958106225323018853885225737147909427851242742472447604799508} a^{7} - \frac{7419802537313710041172452633608588886396982583207323509195241635676023527259534720510380246569}{18507434756313160484879413225659316527020751076729512950752457159698092837475808241492015998360} a^{6} - \frac{42334528750563997660556609371694790916865381484436890428985994245884468954997889232430420119291}{111044608537878962909276479353955899162124506460377077704514742958188557024854849448952095990160} a^{5} - \frac{46623391495121838606908772102849546732253128547127526854947736240257278422239693487664161443997}{111044608537878962909276479353955899162124506460377077704514742958188557024854849448952095990160} a^{4} + \frac{4329486975757649160602985319847096263624824082021178073821880534319096575618760300357071490191}{18507434756313160484879413225659316527020751076729512950752457159698092837475808241492015998360} a^{3} + \frac{40366153503130921071769749961020868792419583534288239562719557089469158206743465870572580734769}{111044608537878962909276479353955899162124506460377077704514742958188557024854849448952095990160} a^{2} + \frac{12172446858786049943939443063842242059417850821590387269031295559016054719041315218000938029879}{111044608537878962909276479353955899162124506460377077704514742958188557024854849448952095990160} a - \frac{638234496951422884908157495022151217867023247649663866944509763517651710122905694413978622233}{11104460853787896290927647935395589916212450646037707770451474295818855702485484944895209599016}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 92771715053800000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

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 $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{149}) \), \(\Q(\sqrt{4321}) \), \(\Q(\sqrt{29}) \), 4.4.95930521.1 x2, 4.4.2781985109.1 x2, \(\Q(\sqrt{29}, \sqrt{149})\), 8.8.7739441146697741881.1

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

Sibling fields

Degree 16 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$149$149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$