↑

Properties

Label 16.0.27668269071...3481.3
Degree $16$
Signature $[0, 8]$
Discriminant $41^{15}\cdot 59^{12}$
Root discriminant $692.03$
Ramified primes $41, 59$
Class number $256$ (GRH)
Class group $[2, 2, 4, 16]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![387255219875, 406935393270, 353322548342, 98354585839, -4193139084, -4522699120, 472182745, 352966659, 111233592, 15872553, 310090, -109940, -17038, -224, 69, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 69*x^14 - 224*x^13 - 17038*x^12 - 109940*x^11 + 310090*x^10 + 15872553*x^9 + 111233592*x^8 + 352966659*x^7 + 472182745*x^6 - 4522699120*x^5 - 4193139084*x^4 + 98354585839*x^3 + 353322548342*x^2 + 406935393270*x + 387255219875)
 
gp: K = bnfinit(x^16 - 4*x^15 + 69*x^14 - 224*x^13 - 17038*x^12 - 109940*x^11 + 310090*x^10 + 15872553*x^9 + 111233592*x^8 + 352966659*x^7 + 472182745*x^6 - 4522699120*x^5 - 4193139084*x^4 + 98354585839*x^3 + 353322548342*x^2 + 406935393270*x + 387255219875, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 69 x^{14} - 224 x^{13} - 17038 x^{12} - 109940 x^{11} + 310090 x^{10} + 15872553 x^{9} + 111233592 x^{8} + 352966659 x^{7} + 472182745 x^{6} - 4522699120 x^{5} - 4193139084 x^{4} + 98354585839 x^{3} + 353322548342 x^{2} + 406935393270 x + 387255219875 \)

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:  \(2766826907141591867309101193151875470463173481=41^{15}\cdot 59^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $692.03$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{59} a^{8} - \frac{2}{59} a^{7} + \frac{3}{59} a^{6} + \frac{12}{59} a^{5} + \frac{26}{59} a^{4} - \frac{25}{59} a^{3} + \frac{29}{59} a^{2} - \frac{22}{59} a - \frac{18}{59}$, $\frac{1}{59} a^{9} - \frac{1}{59} a^{7} + \frac{18}{59} a^{6} - \frac{9}{59} a^{5} + \frac{27}{59} a^{4} - \frac{21}{59} a^{3} - \frac{23}{59} a^{2} - \frac{3}{59} a + \frac{23}{59}$, $\frac{1}{59} a^{10} + \frac{16}{59} a^{7} - \frac{6}{59} a^{6} - \frac{20}{59} a^{5} + \frac{5}{59} a^{4} + \frac{11}{59} a^{3} + \frac{26}{59} a^{2} + \frac{1}{59} a - \frac{18}{59}$, $\frac{1}{59} a^{11} + \frac{26}{59} a^{7} - \frac{9}{59} a^{6} - \frac{10}{59} a^{5} + \frac{8}{59} a^{4} + \frac{13}{59} a^{3} + \frac{9}{59} a^{2} - \frac{20}{59} a - \frac{7}{59}$, $\frac{1}{295} a^{12} + \frac{2}{295} a^{11} - \frac{1}{295} a^{8} - \frac{21}{295} a^{7} - \frac{10}{59} a^{6} + \frac{18}{295} a^{5} + \frac{94}{295} a^{4} + \frac{24}{59} a^{3} - \frac{136}{295} a^{2} - \frac{102}{295} a$, $\frac{1}{295} a^{13} + \frac{1}{295} a^{11} - \frac{1}{295} a^{9} + \frac{1}{295} a^{8} + \frac{82}{295} a^{7} + \frac{133}{295} a^{6} - \frac{47}{295} a^{5} - \frac{98}{295} a^{4} + \frac{74}{295} a^{3} - \frac{18}{59} a^{2} - \frac{41}{295} a - \frac{20}{59}$, $\frac{1}{295} a^{14} - \frac{2}{295} a^{11} - \frac{1}{295} a^{10} + \frac{1}{295} a^{9} - \frac{2}{295} a^{8} + \frac{29}{295} a^{7} + \frac{43}{295} a^{6} + \frac{44}{295} a^{5} + \frac{26}{59} a^{4} + \frac{29}{59} a^{3} - \frac{2}{59} a^{2} + \frac{102}{295} a + \frac{11}{59}$, $\frac{1}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{15} - \frac{233783159556886832538981824458703855020868318879741216439484868964156248}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{14} + \frac{176559521929847801230395714427866266551768181024289623920855677789994711}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{13} - \frac{60695027786450510811270578220643987744483392324840631645643764564812710}{87165472425753654643226048816170181821998765748963506955932353296034220107} a^{12} + \frac{9025163654295131730007969460747879338435121705028577209922185451470105}{87165472425753654643226048816170181821998765748963506955932353296034220107} a^{11} - \frac{820553642901454615991876272694493344775871641190049367598798730381384851}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{10} - \frac{2671359983542356052448678552604582454529949107027394211134105516538662886}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{9} + \frac{3323975951444086958289468516267905381097122316831084296801568549894372214}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{8} - \frac{182780402967159355387240466599928545457696949167563875343783878517869579229}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{7} + \frac{85333982778137856334675607129035028367346213810754695704220221855292595728}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{6} - \frac{208265962361148124526476712796714713732038752366192578139701656064226147583}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{5} + \frac{23949440135562003987052868520347503094778921712791827550314205210392409462}{87165472425753654643226048816170181821998765748963506955932353296034220107} a^{4} + \frac{72630464399608509013816991632636307348336021862455207806282004601121752164}{435827362128768273216130244080850909109993828744817534779661766480171100535} a^{3} + \frac{4054454268032904428578751848483319173721836444822994407879617268714110529}{87165472425753654643226048816170181821998765748963506955932353296034220107} a^{2} - \frac{114027293545436753363134813708864879853808978180870548123302842438409366681}{435827362128768273216130244080850909109993828744817534779661766480171100535} a + \frac{8210883344537386666501431904690813153585036982460897455590623259055159483}{87165472425753654643226048816170181821998765748963506955932353296034220107}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{16}$, which has order $256$ (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:  \( 152478478179000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-2419}) \), 4.0.239914001.1, 8.0.2359907842908948041.5

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ R

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
41Data not computed
$59$59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$