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Properties

Label 16.0.25983016869...0176.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{30}\cdot 193^{8}\cdot 257^{10}$
Root discriminant $1634.62$
Ramified primes $2, 193, 257$
Class number $49751101440$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 12, 32390040]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1082146816, 0, 708784500408, 0, 1590760453744, 0, 279518044450, 0, 9850573750, 0, 122467295, 0, 640747, 0, 1413, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1413*x^14 + 640747*x^12 + 122467295*x^10 + 9850573750*x^8 + 279518044450*x^6 + 1590760453744*x^4 + 708784500408*x^2 + 1082146816)
 
gp: K = bnfinit(x^16 + 1413*x^14 + 640747*x^12 + 122467295*x^10 + 9850573750*x^8 + 279518044450*x^6 + 1590760453744*x^4 + 708784500408*x^2 + 1082146816, 1)
 

Normalized defining polynomial

\( x^{16} + 1413 x^{14} + 640747 x^{12} + 122467295 x^{10} + 9850573750 x^{8} + 279518044450 x^{6} + 1590760453744 x^{4} + 708784500408 x^{2} + 1082146816 \)

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:  \(2598301686999672224815741132125750417700317797810176=2^{30}\cdot 193^{8}\cdot 257^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1634.62$
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:  $2, 193, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{3084} a^{12} - \frac{643}{3084} a^{10} + \frac{101}{1028} a^{8} - \frac{401}{3084} a^{6} + \frac{203}{514} a^{4} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{49344} a^{13} + \frac{899}{49344} a^{11} - \frac{1}{4} a^{10} + \frac{615}{16448} a^{9} + \frac{1141}{49344} a^{7} - \frac{1}{4} a^{6} + \frac{2029}{4112} a^{5} - \frac{1}{2} a^{4} + \frac{5}{96} a^{3} - \frac{1}{2} a^{2} - \frac{17}{48} a$, $\frac{1}{129350059003195208281875876906314798892790656} a^{14} + \frac{9230067908733618995189222679201238117715}{129350059003195208281875876906314798892790656} a^{12} + \frac{2834861377113741446452233523830155619222183}{43116686334398402760625292302104932964263552} a^{10} - \frac{732574084812683906059185717347457857173777}{11759096273017746207443261536937708990253696} a^{8} + \frac{1408834906868355863636909693986572520424405}{10779171583599600690156323075526233241065888} a^{6} + \frac{30541666190163878815609818428734368695130133}{64675029501597604140937938453157399446395328} a^{4} + \frac{34364991085388284983871401867031056400015}{125826905645131525566027117613146691529952} a^{2} + \frac{311636635636233251916176726714247869726}{1310696933803453391312782475136944703437}$, $\frac{1}{258700118006390416563751753812629597785581312} a^{15} + \frac{1365886305912898647312527828379569897093}{258700118006390416563751753812629597785581312} a^{13} - \frac{10300943293464468441284472441992304198623431}{86233372668796805521250584604209865928527104} a^{11} - \frac{1}{4} a^{10} + \frac{3827936319223077466586763051142671377312381}{23518192546035492414886523073875417980507392} a^{9} + \frac{17490921987666372443020419197131240995826025}{43116686334398402760625292302104932964263552} a^{7} - \frac{1}{4} a^{6} - \frac{521851140977966558503126232011220776326767}{129350059003195208281875876906314798892790656} a^{5} - \frac{1}{2} a^{4} + \frac{19404497475225561724298305886637557903359}{62913452822565762783013558806573345764976} a^{3} - \frac{1}{2} a^{2} - \frac{14674267836872068839032909499626204635939}{41942301881710508522009039204382230509984} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{32390040}$, which has order $49751101440$ (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:  \( 22245061053.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{257}) \), \(\Q(\sqrt{193}) \), \(\Q(\sqrt{49601}) \), 4.4.19682073608.1, 4.4.528392.1, \(\Q(\sqrt{193}, \sqrt{257})\), 8.8.387384021510730137664.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.11.15$x^{4} + 30$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
193Data not computed
257Data not computed