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Properties

Label 16.8.19925537421...0896.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{14}\cdot 13^{6}\cdot 29^{8}\cdot 107^{8}\cdot 139^{8}$
Root discriminant $50{,}840.89$
Ramified primes $2, 3, 13, 29, 107, 139$
Class number $2048$ (GRH)
Class group $[2, 2, 4, 4, 4, 8]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1551365056544532363020601, 0, -400049393316527065111416, 0, -713096463830582806866, 0, 320123065218610248, 0, 937154922830376, 0, 105147547800, 0, -203643906, 0, -8232, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8232*x^14 - 203643906*x^12 + 105147547800*x^10 + 937154922830376*x^8 + 320123065218610248*x^6 - 713096463830582806866*x^4 - 400049393316527065111416*x^2 + 1551365056544532363020601)
 
gp: K = bnfinit(x^16 - 8232*x^14 - 203643906*x^12 + 105147547800*x^10 + 937154922830376*x^8 + 320123065218610248*x^6 - 713096463830582806866*x^4 - 400049393316527065111416*x^2 + 1551365056544532363020601, 1)
 

Normalized defining polynomial

\( x^{16} - 8232 x^{14} - 203643906 x^{12} + 105147547800 x^{10} + 937154922830376 x^{8} + 320123065218610248 x^{6} - 713096463830582806866 x^{4} - 400049393316527065111416 x^{2} + 1551365056544532363020601 \)

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:  \(1992553742188905474600208423051202739081327305267455857076982769849672400896=2^{56}\cdot 3^{14}\cdot 13^{6}\cdot 29^{8}\cdot 107^{8}\cdot 139^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50{,}840.89$
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:  $2, 3, 13, 29, 107, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{99} a^{13} + \frac{1}{11} a^{11} + \frac{1}{33} a^{9} + \frac{16}{33} a^{7} - \frac{2}{33} a^{5} - \frac{5}{11} a$, $\frac{1}{4821386133206746253586199630189681861831846265630904723761389584870838151631179049} a^{14} - \frac{55350835372512283552836929185306366315414343796170811574958647401712465506442577}{4821386133206746253586199630189681861831846265630904723761389584870838151631179049} a^{12} - \frac{78661063836596563813027474625026794872225894537289302728236478132248167081636394}{535709570356305139287355514465520206870205140625656080417932176096759794625686561} a^{10} + \frac{31873367410340849803728589846919518424954599316825780534641768670623989937307137}{1607128711068915417862066543396560620610615421876968241253796528290279383877059683} a^{8} - \frac{680511040805876696994877190558657792692930319718798289964863627661358592949947756}{1607128711068915417862066543396560620610615421876968241253796528290279383877059683} a^{6} + \frac{42345261335917787594845011963788196770390568274562209577910627914428947124394831}{146102610097174128896551503945141874600965038352451658295799684390025398534278153} a^{4} + \frac{115261046644465434040266365809495316894021607232727715798258617020396861174266536}{535709570356305139287355514465520206870205140625656080417932176096759794625686561} a^{2} + \frac{80627101326816205220791240870012787378194832878000338281948650191881265872667}{340565524702037596495458051154176863871713376112940928428437492750641954625357}$, $\frac{1}{13998181646363023000444804045468229205485356077001724101619740101619278288844426998382554831} a^{15} + \frac{17712439854474754167506891048318592971056443382606636436248398681597196781275500630949441}{4666060548787674333481601348489409735161785359000574700539913367206426096281475666127518277} a^{13} + \frac{9775525196342026293730605115476112795054691243727301201781630686992989221046258499825157}{4666060548787674333481601348489409735161785359000574700539913367206426096281475666127518277} a^{11} + \frac{608343042328851326676923271860046177946016279284943840914862867610481481600513787695816455}{4666060548787674333481601348489409735161785359000574700539913367206426096281475666127518277} a^{9} + \frac{1866416203021997625199442960960093856190210410334511381362065873951609187327093049255332522}{4666060548787674333481601348489409735161785359000574700539913367206426096281475666127518277} a^{7} + \frac{70289466097833626697970104493820556348278370493832465925193860297504876367440759636228692}{141395774205687101014593980257254840459448041181835596986057980824437154432771989882652069} a^{5} - \frac{257453793041816811875318729580582513084028574820529257441359590267347998433301827999502748}{1555353516262558111160533782829803245053928453000191566846637789068808698760491888709172759} a^{3} + \frac{1803940182085036387615113465546128816196899494338090814882163270410464185527458829237759}{10876598015822084693430306173634987727649849321679661306619844678802858033290153067896313} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{8}$, which has order $2048$ (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:  \( 81753725747100000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.1393025246413632.1, 4.4.179712.1, 4.4.2571738916455936.1, 8.8.4470956552783831393152593634000896.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
3Data not computed
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$107$107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.8.4.1$x^{8} + 45796 x^{4} - 1225043 x^{2} + 524318404$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$139$139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$