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Properties

Label 16.0.12436160298...7041.5
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 53^{14}$
Root discriminant $760.18$
Ramified primes $37, 53$
Class number $76527504$ (GRH)
Class group $[2, 4374, 8748]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![367571482355344, 173922414925936, 5952716972260, 5667181894124, 3345800137369, 125714438271, 58434692944, 3574509683, 872048656, 19722151, 5855078, 258957, 20832, 361, 204, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 204*x^14 + 361*x^13 + 20832*x^12 + 258957*x^11 + 5855078*x^10 + 19722151*x^9 + 872048656*x^8 + 3574509683*x^7 + 58434692944*x^6 + 125714438271*x^5 + 3345800137369*x^4 + 5667181894124*x^3 + 5952716972260*x^2 + 173922414925936*x + 367571482355344)
 
gp: K = bnfinit(x^16 - 7*x^15 + 204*x^14 + 361*x^13 + 20832*x^12 + 258957*x^11 + 5855078*x^10 + 19722151*x^9 + 872048656*x^8 + 3574509683*x^7 + 58434692944*x^6 + 125714438271*x^5 + 3345800137369*x^4 + 5667181894124*x^3 + 5952716972260*x^2 + 173922414925936*x + 367571482355344, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 204 x^{14} + 361 x^{13} + 20832 x^{12} + 258957 x^{11} + 5855078 x^{10} + 19722151 x^{9} + 872048656 x^{8} + 3574509683 x^{7} + 58434692944 x^{6} + 125714438271 x^{5} + 3345800137369 x^{4} + 5667181894124 x^{3} + 5952716972260 x^{2} + 173922414925936 x + 367571482355344 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12436160298838170417198712919084997051554477041=37^{14}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $760.18$
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:  $37, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{8} a^{8} - \frac{1}{12} a^{7} - \frac{5}{24} a^{4} - \frac{7}{24} a^{3} + \frac{7}{24} a^{2} + \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{240} a^{11} - \frac{1}{240} a^{10} - \frac{9}{80} a^{9} - \frac{13}{120} a^{8} - \frac{1}{20} a^{7} - \frac{1}{4} a^{6} + \frac{7}{240} a^{5} - \frac{43}{240} a^{4} + \frac{67}{240} a^{3} + \frac{23}{120} a^{2} + \frac{1}{60} a + \frac{1}{5}$, $\frac{1}{7920} a^{12} + \frac{7}{3960} a^{11} + \frac{59}{3960} a^{10} - \frac{277}{2640} a^{9} - \frac{49}{440} a^{8} - \frac{13}{99} a^{7} + \frac{137}{720} a^{6} - \frac{329}{3960} a^{5} + \frac{631}{3960} a^{4} + \frac{107}{240} a^{3} + \frac{1027}{3960} a^{2} + \frac{427}{1980} a - \frac{49}{99}$, $\frac{1}{7920} a^{13} - \frac{1}{660} a^{11} + \frac{91}{7920} a^{10} + \frac{13}{264} a^{9} - \frac{7}{180} a^{8} - \frac{377}{1584} a^{7} - \frac{163}{660} a^{6} - \frac{59}{495} a^{5} + \frac{301}{1584} a^{4} - \frac{29}{3960} a^{3} + \frac{17}{330} a^{2} + \frac{349}{990} a - \frac{167}{495}$, $\frac{1}{237600} a^{14} - \frac{1}{23760} a^{13} + \frac{1}{47520} a^{12} - \frac{61}{59400} a^{11} - \frac{2441}{237600} a^{10} + \frac{259}{59400} a^{9} + \frac{12139}{237600} a^{8} - \frac{1949}{7920} a^{7} + \frac{763}{5280} a^{6} + \frac{6647}{59400} a^{5} + \frac{469}{9504} a^{4} - \frac{23909}{59400} a^{3} - \frac{2677}{9900} a^{2} + \frac{12989}{29700} a + \frac{2828}{7425}$, $\frac{1}{259712095157487663872171453935777461042264436465018866659601069245918984128570377880800} a^{15} + \frac{6581458246786443874511982099498005767061064930256063834117842380933101351034699}{5194241903149753277443429078715549220845288729300377333192021384918379682571407557616} a^{14} - \frac{991222205452156716430466864084650120867748644287696426494342357751597524874303857}{17314139677165844258144763595718497402817629097667924443973404616394598941904691858720} a^{13} - \frac{2336512157807472389469158185808923773810797927365358842836210824763524480274654649}{43285349192914610645361908989296243507044072744169811109933511540986497354761729646800} a^{12} + \frac{52078855332407299875135942756557717823738691760328677862430498890859449736871849039}{259712095157487663872171453935777461042264436465018866659601069245918984128570377880800} a^{11} - \frac{185319264027159398560825437365035023979732739529755937822096917823320412389387699}{10465509959602178589304136602827911873076419909131965935670578225577006130261540050} a^{10} - \frac{3522519419208061665350168534308260502100816357638893472071802346308498902378418395817}{86570698385829221290723817978592487014088145488339622219867023081972994709523459293600} a^{9} - \frac{440511217086100293337517618515046180693092330866704422219368139295337403841594952917}{5194241903149753277443429078715549220845288729300377333192021384918379682571407557616} a^{8} - \frac{210625976198351298461429839949655924674587414207949393484599830961022388014439923731}{17314139677165844258144763595718497402817629097667924443973404616394598941904691858720} a^{7} - \frac{7415442945916742023544337840992864150520530206402066544271489577930651853420282570501}{129856047578743831936085726967888730521132218232509433329800534622959492064285188940400} a^{6} + \frac{187224201558942057423475794554113434278866888364360172381662137938214701575803010371}{2733811527973554356549173199323973274129099331210724912206327044693884043458635556640} a^{5} + \frac{70986324843418850731557765328160266889685255880599057932751692537422362994161654401}{300592702728573685037235479092335024354472727390068132707871607923517342741400900325} a^{4} + \frac{642148140072555376506449020160453543646465719782623844317468300982042267720378881811}{129856047578743831936085726967888730521132218232509433329800534622959492064285188940400} a^{3} - \frac{11291476925066108500166038244137367954168183153231351683134325296452320981454921277587}{64928023789371915968042863483944365260566109116254716664900267311479746032142594470200} a^{2} + \frac{2316060648655559274262673330885811376987947724031929813726932232996433829130896851729}{10821337298228652661340477247324060876761018186042452777483377885246624338690432411700} a - \frac{525727719958013537480124576080514227781004004149800944027800057699805209931300535367}{1623200594734297899201071587098609131514152727906367916622506682786993650803564861755}$

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

Class group and class number

$C_{2}\times C_{4374}\times C_{8748}$, which has order $76527504$ (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:  $7$
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:  \( 4687632840.89 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{1961}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{37}) \), 4.4.7541066681.1, 4.4.7541066681.2, \(\Q(\sqrt{37}, \sqrt{53})\), 8.0.111517533593772465647321.1 x2, 8.0.111517533593772465647321.2 x2, 8.8.56867686687288355761.3

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ 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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$