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Properties

Label 16.8.80291031759...9473.4
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 73^{15}$
Root discriminant $1315.35$
Ramified primes $37, 73$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31014141249191103, -18957448548520857, 2681157447026307, 608115581856810, -75494837260791, -3976708155048, -109039674744, -35610842313, 7467374561, 134162437, -7413755, 1330723, -185649, 119, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 119*x^13 - 185649*x^12 + 1330723*x^11 - 7413755*x^10 + 134162437*x^9 + 7467374561*x^8 - 35610842313*x^7 - 109039674744*x^6 - 3976708155048*x^5 - 75494837260791*x^4 + 608115581856810*x^3 + 2681157447026307*x^2 - 18957448548520857*x + 31014141249191103)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 119*x^13 - 185649*x^12 + 1330723*x^11 - 7413755*x^10 + 134162437*x^9 + 7467374561*x^8 - 35610842313*x^7 - 109039674744*x^6 - 3976708155048*x^5 - 75494837260791*x^4 + 608115581856810*x^3 + 2681157447026307*x^2 - 18957448548520857*x + 31014141249191103, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} + 119 x^{13} - 185649 x^{12} + 1330723 x^{11} - 7413755 x^{10} + 134162437 x^{9} + 7467374561 x^{8} - 35610842313 x^{7} - 109039674744 x^{6} - 3976708155048 x^{5} - 75494837260791 x^{4} + 608115581856810 x^{3} + 2681157447026307 x^{2} - 18957448548520857 x + 31014141249191103 \)

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:  \(80291031759964036646878404141998486437104406229473=37^{14}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1315.35$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{4}{9} a^{8} - \frac{4}{9} a^{7} + \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{4}{27} a^{9} - \frac{13}{27} a^{8} - \frac{8}{27} a^{6} - \frac{11}{27} a^{5} + \frac{7}{27} a^{4} + \frac{8}{27} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{227286} a^{12} - \frac{1277}{113643} a^{11} + \frac{10513}{227286} a^{10} - \frac{17420}{113643} a^{9} + \frac{7870}{37881} a^{8} + \frac{3038}{113643} a^{7} - \frac{27256}{113643} a^{6} + \frac{84187}{227286} a^{5} + \frac{80651}{227286} a^{4} - \frac{13277}{75762} a^{3} - \frac{5021}{25254} a^{2} + \frac{86}{183} a + \frac{309}{2806}$, $\frac{1}{12955302} a^{13} - \frac{11}{6477651} a^{12} - \frac{5107}{563274} a^{11} - \frac{143090}{6477651} a^{10} - \frac{76753}{2159217} a^{9} - \frac{2593822}{6477651} a^{8} + \frac{2702549}{6477651} a^{7} - \frac{2048423}{12955302} a^{6} + \frac{4341047}{12955302} a^{5} + \frac{25753}{159942} a^{4} + \frac{2401}{159942} a^{3} + \frac{116692}{239913} a^{2} + \frac{22481}{159942} a - \frac{595}{1403}$, $\frac{1}{205438158028947006} a^{14} + \frac{7832199497}{205438158028947006} a^{13} + \frac{39084244784}{102719079014473503} a^{12} - \frac{1109821385604481}{205438158028947006} a^{11} + \frac{154872902862953}{2977364609115174} a^{10} + \frac{4037263918212263}{102719079014473503} a^{9} + \frac{14828291682974876}{102719079014473503} a^{8} + \frac{51450201998247415}{205438158028947006} a^{7} - \frac{22034590033093571}{102719079014473503} a^{6} + \frac{368017439239439}{22826462003216334} a^{5} + \frac{3631545328004999}{22826462003216334} a^{4} - \frac{7393405998103}{200232122835231} a^{3} - \frac{310568034777889}{1268136777956463} a^{2} - \frac{337588367880439}{845424518637642} a - \frac{5715515126707}{14832009098906}$, $\frac{1}{3286653989753598213525212953135904316762640457540776615458002691098056206309051004998} a^{15} + \frac{1076797454948807066090128612346523197556528396179879015863813848109}{1643326994876799106762606476567952158381320228770388307729001345549028103154525502499} a^{14} + \frac{30824402033048425615794917033137061572227369387264149017295208533779843862731}{3286653989753598213525212953135904316762640457540776615458002691098056206309051004998} a^{13} - \frac{1826804882505063673532079961999588783670359673136690906999543969256182862130300}{1643326994876799106762606476567952158381320228770388307729001345549028103154525502499} a^{12} + \frac{841551479339608399454880254355861727609720699583065662563441354513993611703942594}{547775664958933035587535492189317386127106742923462769243000448516342701051508500833} a^{11} - \frac{75700387011304929558691269162438250004080231653272873066883609543035287743881246902}{1643326994876799106762606476567952158381320228770388307729001345549028103154525502499} a^{10} - \frac{14405706307972406589103723061184512877324862674149825439529337370101716826637681457}{86490894467199952987505604029892218862174748882652016196263228713106742271290815921} a^{9} + \frac{1143553610189907344754474071174604544997449271089806630542629671191384897883001545637}{3286653989753598213525212953135904316762640457540776615458002691098056206309051004998} a^{8} + \frac{820826168878642754632193409888019935375388214537504620555107643073331419956941763373}{3286653989753598213525212953135904316762640457540776615458002691098056206309051004998} a^{7} - \frac{89140028910485039544993615405969897026410875349844412146770875224657423841584867795}{365183776639288690391690328126211590751404495282308512828666965677561800701005667222} a^{6} - \frac{19813794110328529816564051221665786429909133177641498122557766697021092254249627479}{121727925546429563463896776042070530250468165094102837609555655225853933567001889074} a^{5} - \frac{302416511386528920823393203078872360361822031393470894822089471789408973048455477}{20287987591071593910649462673678421708411360849017139601592609204308988927833648179} a^{4} - \frac{17902293135203626301998350241789091853848777916939538405108274954837173466897417651}{40575975182143187821298925347356843416822721698034279203185218408617977855667296358} a^{3} - \frac{535080822552761533361095240730501383485736347085552659079231733558354465425481498}{6762662530357197970216487557892807236137120283005713200530869734769662975944549393} a^{2} - \frac{640341284920102087745046113582913715221962831181214883737401180760947424894319842}{2254220843452399323405495852630935745379040094335237733510289911589887658648183131} a - \frac{17843262277491275690796586015592030639303509273838183311715133887806094949899361}{39547734095656128480798172853174311322439299900618205851057717747191011555231283}$

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:  \( 3883759568770000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.532564273.1, 8.8.28344602131194663732673.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ $16$ $16$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
$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}$
73Data not computed