↑

Properties

Label 16.0.96825103549...7184.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{14}\cdot 11^{14}\cdot 2111^{4}$
Root discriminant $485.99$
Ramified primes $2, 3, 11, 2111$
Class number $6536829696$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 51068982]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T493)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2940120561720576, 405991720954560, 358906331026404, 37916090753304, 25049349509778, 1485492313944, 891724768485, 33103102647, 16054876969, 375878024, 144386740, 1514696, 646954, 1304, 1333, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 1333*x^14 + 1304*x^13 + 646954*x^12 + 1514696*x^11 + 144386740*x^10 + 375878024*x^9 + 16054876969*x^8 + 33103102647*x^7 + 891724768485*x^6 + 1485492313944*x^5 + 25049349509778*x^4 + 37916090753304*x^3 + 358906331026404*x^2 + 405991720954560*x + 2940120561720576)
 
gp: K = bnfinit(x^16 - x^15 + 1333*x^14 + 1304*x^13 + 646954*x^12 + 1514696*x^11 + 144386740*x^10 + 375878024*x^9 + 16054876969*x^8 + 33103102647*x^7 + 891724768485*x^6 + 1485492313944*x^5 + 25049349509778*x^4 + 37916090753304*x^3 + 358906331026404*x^2 + 405991720954560*x + 2940120561720576, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 1333 x^{14} + 1304 x^{13} + 646954 x^{12} + 1514696 x^{11} + 144386740 x^{10} + 375878024 x^{9} + 16054876969 x^{8} + 33103102647 x^{7} + 891724768485 x^{6} + 1485492313944 x^{5} + 25049349509778 x^{4} + 37916090753304 x^{3} + 358906331026404 x^{2} + 405991720954560 x + 2940120561720576 \)

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:  \(9682510354914354784154938551289076131037184=2^{28}\cdot 3^{14}\cdot 11^{14}\cdot 2111^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $485.99$
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:  $2, 3, 11, 2111$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{214253416033073418855931200637952266654301433131985348916132281509769688762770978758646169554603612717147893632} a^{15} + \frac{637938536372088140024046063699839395573605122116082387190181874926852731799710522110375502276741484494829129}{6911400517195916737288103246385556988848433326838237061810718758379667379444225121246650630793664926359609472} a^{14} + \frac{444179325800146247061640486467091061843161765419551396691878846277406281085885928730517833684814340074299331}{6911400517195916737288103246385556988848433326838237061810718758379667379444225121246650630793664926359609472} a^{13} + \frac{1384183859127338161234746937436216741857651322786209997012420451773231333910410341442898248469454048930639345}{13390838502067088678495700039872016665893839570749084307258267594360605547673186172415385597162725794821743352} a^{12} - \frac{39408531195806801158698326620602956162366967425334285832303520664908211502793443932010159323980168713879843947}{107126708016536709427965600318976133327150716565992674458066140754884844381385489379323084777301806358573946816} a^{11} + \frac{10566603437299083535950476684153191416269653899639683639871683420904897881494757200630114872019766698221955351}{26781677004134177356991400079744033331787679141498168614516535188721211095346372344830771194325451589643486704} a^{10} + \frac{24529405276524453003410406512777764777885908676662072843280261221135837841379154673681581277022922431897305981}{53563354008268354713982800159488066663575358282996337229033070377442422190692744689661542388650903179286973408} a^{9} + \frac{12115308571602109521088007957729304068026041678022307547628352579600520093780114064580668574320382101760155421}{26781677004134177356991400079744033331787679141498168614516535188721211095346372344830771194325451589643486704} a^{8} - \frac{76832737619223259886271830192662418310030847244124155760819900751298489285530591650057961361025425734163781783}{214253416033073418855931200637952266654301433131985348916132281509769688762770978758646169554603612717147893632} a^{7} - \frac{26349032507024760679489262832398678455210246806483566758294217889816592737569508794907251590714854939827694641}{214253416033073418855931200637952266654301433131985348916132281509769688762770978758646169554603612717147893632} a^{6} + \frac{3291737429903049400212954967021967901249931306991277706985795783264951344383422855324984177809067188938196749}{214253416033073418855931200637952266654301433131985348916132281509769688762770978758646169554603612717147893632} a^{5} - \frac{1149955504727939275016178699398056356950657712241680164445831964998077831320219180237159674006960343033853547}{13390838502067088678495700039872016665893839570749084307258267594360605547673186172415385597162725794821743352} a^{4} - \frac{45022939929165336778572107118211981204169690136526650509308563600338985149772624071854484896232680492775249719}{107126708016536709427965600318976133327150716565992674458066140754884844381385489379323084777301806358573946816} a^{3} + \frac{9907696774362029827636180922553920411905100312694067770871606435277342551456402395662059476666404487757155625}{26781677004134177356991400079744033331787679141498168614516535188721211095346372344830771194325451589643486704} a^{2} + \frac{3881637164224963352467390317700706620052029396721270504494824086163930090536854061683473086497657860673039721}{53563354008268354713982800159488066663575358282996337229033070377442422190692744689661542388650903179286973408} a + \frac{170167425161694055700406375812386918694049841042711638436932826887608142286841600694612103560190865223611037}{515032250079503410711373078456616025611301521951887857971471830552330982602814852785207138352412530570067052}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{51068982}$, which has order $6536829696$ (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:  \( 680881.531768 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T493):

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

\(\Q(\sqrt{33}) \), 4.4.95832.1, 4.4.287496.1, 4.4.13068.1, 8.8.330615800064.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.24.22$x^{8} + 4 x^{4} + 36$$8$$1$$24$$D_4\times C_2$$[2, 3, 4]^{2}$
$3$3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
11Data not computed
2111Data not computed