↑

Properties

Label 16.16.2625581163...3841.1
Degree $16$
Signature $[16, 0]$
Discriminant $61^{14}\cdot 149^{14}$
Root discriminant $2908.72$
Ramified primes $61, 149$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-35542411485577782059, 17337036358118728404, -1649219797768775897, -300399178945081685, 43382677650078083, 2024106400885404, -382817796251067, -7299192869405, 1683967369007, 16436844988, -4058995913, -23875039, 5392416, 19982, -3676, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 3676*x^14 + 19982*x^13 + 5392416*x^12 - 23875039*x^11 - 4058995913*x^10 + 16436844988*x^9 + 1683967369007*x^8 - 7299192869405*x^7 - 382817796251067*x^6 + 2024106400885404*x^5 + 43382677650078083*x^4 - 300399178945081685*x^3 - 1649219797768775897*x^2 + 17337036358118728404*x - 35542411485577782059)
 
gp: K = bnfinit(x^16 - 7*x^15 - 3676*x^14 + 19982*x^13 + 5392416*x^12 - 23875039*x^11 - 4058995913*x^10 + 16436844988*x^9 + 1683967369007*x^8 - 7299192869405*x^7 - 382817796251067*x^6 + 2024106400885404*x^5 + 43382677650078083*x^4 - 300399178945081685*x^3 - 1649219797768775897*x^2 + 17337036358118728404*x - 35542411485577782059, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 3676 x^{14} + 19982 x^{13} + 5392416 x^{12} - 23875039 x^{11} - 4058995913 x^{10} + 16436844988 x^{9} + 1683967369007 x^{8} - 7299192869405 x^{7} - 382817796251067 x^{6} + 2024106400885404 x^{5} + 43382677650078083 x^{4} - 300399178945081685 x^{3} - 1649219797768775897 x^{2} + 17337036358118728404 x - 35542411485577782059 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26255811630589850827051927649928110197766220114319683841=61^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2908.72$
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:  $61, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{4}{19} a^{12} - \frac{9}{19} a^{11} + \frac{7}{19} a^{10} + \frac{5}{19} a^{9} - \frac{9}{19} a^{8} + \frac{6}{19} a^{7} + \frac{1}{19} a^{6} - \frac{1}{19} a^{5} + \frac{2}{19} a^{4} - \frac{6}{19} a^{3} - \frac{8}{19} a^{2} + \frac{7}{19} a$, $\frac{1}{10849} a^{14} + \frac{93}{10849} a^{13} + \frac{4391}{10849} a^{12} + \frac{1813}{10849} a^{11} + \frac{16}{571} a^{10} + \frac{3212}{10849} a^{9} - \frac{5256}{10849} a^{8} - \frac{1773}{10849} a^{7} - \frac{1158}{10849} a^{6} + \frac{154}{571} a^{5} - \frac{40}{10849} a^{4} + \frac{4293}{10849} a^{3} - \frac{1396}{10849} a^{2} - \frac{765}{10849} a - \frac{126}{571}$, $\frac{1}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{15} - \frac{2595612697919574054824542207049293473501756394812104720299060113856578693173772331571467103976786870511681944449862}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{14} - \frac{137301199000820469415936297536486018063699527992857698441185086083795055223459425602929013799738529376297123141674668}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{13} - \frac{6406603078189528170775545903151291373449144108432806183933692143687987131654334211082195361791393185448160937762058230}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{12} - \frac{18712064237336460256334856949192526123576019692783298752105603328008090087001052227687840434984866534719217862180281061}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{11} + \frac{22333555040147323722746446625034207527514103641945749580782436504561413383409026848709468155488850413881108341174807388}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{10} + \frac{30126122040185758712242189921366386625072113608880117397837204115410704795734496913040141204132124324563435495030977896}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{9} - \frac{10035029751255546166712624799167764732642035810494951071256150852428503655208213593666702493972918432213054733306963006}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{8} - \frac{3613481441177662908332748610914069417969697126395020070417401787455311519784724138326459868255342271642894301934611298}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{7} + \frac{18148182464525383156714681450961650512746628681977493729944007675835377385135297882768149876276787015369328407942631228}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{6} - \frac{16528891662912774208546529284240038786144943319083039656348376239834574582715044860622795397049521706633421341780159834}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{5} + \frac{15491786770615111286996771697048684856502192922096132196096110035382067846436525849473612664616707206060365418734686978}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{4} + \frac{15078738900051854691730586381054658525068145787264378722814737597197078159539839822715883287242016048308463892411209396}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{3} + \frac{29207876028938169798661806860436444174530546897569265793168905273634446785599220303539819741885372768720936517533273254}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a^{2} + \frac{11893799401668569218710487766521620556895879782775059824014783077749228783464666258786408391492803884149113712841123488}{62617839997947811781944638876087237896012593613181556489871291765722486169297358318595153530962300961164534881136879651} a + \frac{1233767406074821041348900896155898754006929369011189571448222761261819938877489967131931251683218211094000983799341984}{3295675789365674304312875730320380941895399663851660867887962724511709798384071490452376501629594787429712362165098929}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 16137194048900000000000 \) (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

\(\Q(\sqrt{9089}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{61}, \sqrt{149})\), 8.8.563763066196879006536961.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{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}$