↑

Properties

Label 16.8.55273437382...6449.2
Degree $16$
Signature $[8, 4]$
Discriminant $37^{10}\cdot 101^{14}$
Root discriminant $541.89$
Ramified primes $37, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-180997979, 902653490, -719056917, 77391304, 15506419, -14124804, 5356541, -1294348, 139258, 80208, -21897, 5644, -863, -126, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 126*x^13 - 863*x^12 + 5644*x^11 - 21897*x^10 + 80208*x^9 + 139258*x^8 - 1294348*x^7 + 5356541*x^6 - 14124804*x^5 + 15506419*x^4 + 77391304*x^3 - 719056917*x^2 + 902653490*x - 180997979)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 126*x^13 - 863*x^12 + 5644*x^11 - 21897*x^10 + 80208*x^9 + 139258*x^8 - 1294348*x^7 + 5356541*x^6 - 14124804*x^5 + 15506419*x^4 + 77391304*x^3 - 719056917*x^2 + 902653490*x - 180997979, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 126 x^{13} - 863 x^{12} + 5644 x^{11} - 21897 x^{10} + 80208 x^{9} + 139258 x^{8} - 1294348 x^{7} + 5356541 x^{6} - 14124804 x^{5} + 15506419 x^{4} + 77391304 x^{3} - 719056917 x^{2} + 902653490 x - 180997979 \)

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:  \(55273437382717328226723918791876367935996449=37^{10}\cdot 101^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $541.89$
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:  $37, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{400} a^{14} - \frac{1}{80} a^{13} - \frac{3}{200} a^{12} + \frac{1}{400} a^{11} + \frac{7}{200} a^{10} - \frac{43}{400} a^{9} - \frac{2}{25} a^{8} - \frac{41}{400} a^{7} + \frac{63}{400} a^{6} - \frac{17}{40} a^{5} - \frac{49}{100} a^{4} - \frac{81}{200} a^{3} + \frac{117}{400} a^{2} - \frac{47}{400} a + \frac{49}{400}$, $\frac{1}{3666183770074837308108817554846127957437044205462800} a^{15} + \frac{1274248317663108412977235577479802218265139236663}{1833091885037418654054408777423063978718522102731400} a^{14} - \frac{78025025703175311732151470791693547654382213155461}{3666183770074837308108817554846127957437044205462800} a^{13} + \frac{1281876520062518109422832944673218317235629256333}{733236754014967461621763510969225591487408841092560} a^{12} - \frac{83558870478108923369520736887376559422313649944471}{733236754014967461621763510969225591487408841092560} a^{11} - \frac{413051153344214571463786086042675365303345310205759}{3666183770074837308108817554846127957437044205462800} a^{10} - \frac{20683646637148636215687949637682733356439451052413}{733236754014967461621763510969225591487408841092560} a^{9} - \frac{83876241721961243091116003025522356135565837839483}{3666183770074837308108817554846127957437044205462800} a^{8} - \frac{152525302413653767452956022912720589354500351132077}{916545942518709327027204388711531989359261051365700} a^{7} + \frac{63003053927165025601385226852813787446559545952433}{3666183770074837308108817554846127957437044205462800} a^{6} + \frac{581096786528430347667354766727727676078561023503067}{1833091885037418654054408777423063978718522102731400} a^{5} - \frac{562528121576077150196384360408307282952141038459069}{1833091885037418654054408777423063978718522102731400} a^{4} - \frac{444061719027172854829252263186892805617005019211}{10327278225562921994672725506608811147709983677360} a^{3} - \frac{10162244850590807369025205161149388251689855334764}{45827297125935466351360219435576599467963052568285} a^{2} + \frac{92548944804894745386738260884709359164650399417949}{458272971259354663513602194355765994679630525682850} a - \frac{895105421185235565786492966104271852169752935559931}{3666183770074837308108817554846127957437044205462800}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 13328976344000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3737}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{37}) \), 4.4.1410482069.1 x2, 4.4.38121137.1 x2, \(\Q(\sqrt{37}, \sqrt{101})\), 8.8.1989459666970520761.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$37$37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.6.2$x^{8} + 333 x^{4} + 34225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.8.7.2$x^{8} - 404$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
101.8.7.2$x^{8} - 404$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$