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Properties

Label 16.0.16622447092...4761.6
Degree $16$
Signature $[0, 8]$
Discriminant $41^{15}\cdot 83^{12}$
Root discriminant $893.91$
Ramified primes $41, 83$
Class number $1280$ (GRH)
Class group $[2, 2, 2, 2, 80]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9312471101944163, 4893603985587031, 1553171376434168, 319122527760339, 49556273455148, 5568200365384, 489495293654, 32214962714, 2277412340, 220167189, 25630993, 1956747, 75130, -2479, -259, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 259*x^14 - 2479*x^13 + 75130*x^12 + 1956747*x^11 + 25630993*x^10 + 220167189*x^9 + 2277412340*x^8 + 32214962714*x^7 + 489495293654*x^6 + 5568200365384*x^5 + 49556273455148*x^4 + 319122527760339*x^3 + 1553171376434168*x^2 + 4893603985587031*x + 9312471101944163)
 
gp: K = bnfinit(x^16 - 4*x^15 - 259*x^14 - 2479*x^13 + 75130*x^12 + 1956747*x^11 + 25630993*x^10 + 220167189*x^9 + 2277412340*x^8 + 32214962714*x^7 + 489495293654*x^6 + 5568200365384*x^5 + 49556273455148*x^4 + 319122527760339*x^3 + 1553171376434168*x^2 + 4893603985587031*x + 9312471101944163, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 259 x^{14} - 2479 x^{13} + 75130 x^{12} + 1956747 x^{11} + 25630993 x^{10} + 220167189 x^{9} + 2277412340 x^{8} + 32214962714 x^{7} + 489495293654 x^{6} + 5568200365384 x^{5} + 49556273455148 x^{4} + 319122527760339 x^{3} + 1553171376434168 x^{2} + 4893603985587031 x + 9312471101944163 \)

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:  \(166224470923824362155644060831037034515503474761=41^{15}\cdot 83^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $893.91$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{15} + \frac{47145367438028563886674107207411411167446882719235069719433706914705000358794673852542539653893306057484916}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{14} + \frac{120201160681282569713670620169472818225399961082057527082395981713383979624211614835161251411677248682394190}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{13} - \frac{76996018986264212324868116285287589961857041628918433120416455718283848555464125828710941947431477312318637}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{12} + \frac{89403970563768338627937631708706687568423940070576536415687310062786980092843067129065176896623452918053613}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{11} + \frac{79555381091620603325373818584217706649912650451514453503805329436181881539448758221422699827845131817676522}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{10} + \frac{146934672437338626332179124737948062539224050165843949141081283861733305614480361793173363474593765379884478}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{9} - \frac{39059916957867688811365940922412865268843826779604062005061165509600467557911916755062922314389924242402056}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{8} + \frac{145496358100252371839878167436695449658833618478330876507195842051406828809734458470229808279626895315419558}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{7} - \frac{65423076254047710741881050590899968054658607840363593915165894363654425273575803217652011289027849548194254}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{6} - \frac{185488334132470910571076576725351831652772979969302143283126318481949893439725595751801913611698608613524431}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{5} + \frac{2461638864200898578085598652198606036401300230590695005746206682499744728566811200068893612847863241514150}{21303994982428163683464371463956274282280438307304374599661831658912495322443914639388466399109410561355657} a^{4} - \frac{10350214970188995325890537423817350706922265974227258011013719129283966190105529702861093890948762508084018}{21303994982428163683464371463956274282280438307304374599661831658912495322443914639388466399109410561355657} a^{3} + \frac{61196853005505394157824114961822884981662563536071985306303435721031218127268057883356759615249834430451809}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a^{2} + \frac{144149282883654015922406129178797661016020772990464866050844569484723330271365723497020580794310970569194073}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483} a + \frac{109508990714498832163411734025462747557927651931401441165183374550704504479717522360423157810030039073246287}{404775904666135109985823057815169211363328327838783117393574801519337411126434378148380861583078800665757483}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{80}$, which has order $1280$ (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:  \( 59776039123500 \) (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

\(\Q(\sqrt{-3403}) \), 4.0.474796769.1, 8.0.9242710845966413801.3

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
41Data not computed
$83$83.8.6.2$x^{8} + 249 x^{4} + 27556$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
83.8.6.2$x^{8} + 249 x^{4} + 27556$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$