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Properties

Label 16.8.70440191930...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{38}\cdot 5^{8}\cdot 7^{8}\cdot 241^{8}$
Root discriminant $476.42$
Ramified primes $2, 5, 7, 241$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-815756591, -1822563328, -837002750, 263821580, 209648745, 119982956, 21793570, -7061852, -2053388, -160984, 75894, 28276, 633, -244, -90, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 90*x^14 - 244*x^13 + 633*x^12 + 28276*x^11 + 75894*x^10 - 160984*x^9 - 2053388*x^8 - 7061852*x^7 + 21793570*x^6 + 119982956*x^5 + 209648745*x^4 + 263821580*x^3 - 837002750*x^2 - 1822563328*x - 815756591)
 
gp: K = bnfinit(x^16 - 4*x^15 - 90*x^14 - 244*x^13 + 633*x^12 + 28276*x^11 + 75894*x^10 - 160984*x^9 - 2053388*x^8 - 7061852*x^7 + 21793570*x^6 + 119982956*x^5 + 209648745*x^4 + 263821580*x^3 - 837002750*x^2 - 1822563328*x - 815756591, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 90 x^{14} - 244 x^{13} + 633 x^{12} + 28276 x^{11} + 75894 x^{10} - 160984 x^{9} - 2053388 x^{8} - 7061852 x^{7} + 21793570 x^{6} + 119982956 x^{5} + 209648745 x^{4} + 263821580 x^{3} - 837002750 x^{2} - 1822563328 x - 815756591 \)

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:  \(7044019193056619128660199158998630400000000=2^{38}\cdot 5^{8}\cdot 7^{8}\cdot 241^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $476.42$
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:  $2, 5, 7, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{5} + \frac{5}{12} a^{4} + \frac{5}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{4} a - \frac{1}{12}$, $\frac{1}{780} a^{14} - \frac{1}{65} a^{13} - \frac{23}{780} a^{12} + \frac{31}{260} a^{11} - \frac{29}{780} a^{10} - \frac{31}{195} a^{9} - \frac{1}{10} a^{8} - \frac{179}{780} a^{7} - \frac{49}{780} a^{6} + \frac{1}{780} a^{5} - \frac{167}{780} a^{4} - \frac{53}{195} a^{3} + \frac{59}{780} a^{2} - \frac{139}{780} a + \frac{1}{30}$, $\frac{1}{23227538611363200781015872865785554109245658666603032898416580} a^{15} - \frac{1053353055683526406488785321430167607515965963982081376841}{23227538611363200781015872865785554109245658666603032898416580} a^{14} + \frac{73704884754565468877029005003873370248810444356933996550041}{2322753861136320078101587286578555410924565866660303289841658} a^{13} - \frac{142697585210462984638727297728285374618475114394839405085473}{1161376930568160039050793643289277705462282933330151644920829} a^{12} + \frac{236630257132034752039119664027456945925572870054341482374983}{7742512870454400260338624288595184703081886222201010966138860} a^{11} - \frac{104713200886940929569551107930229025001316714556072454481977}{5806884652840800195253968216446388527311414666650758224604145} a^{10} - \frac{3489140244401616603233388198646074308428387480351630266531887}{23227538611363200781015872865785554109245658666603032898416580} a^{9} - \frac{1931335160324895805142063567521331934482414402778223702129359}{7742512870454400260338624288595184703081886222201010966138860} a^{8} - \frac{190839916582047246109664038673902806756984043913565722519197}{5806884652840800195253968216446388527311414666650758224604145} a^{7} - \frac{556380199321288800063562952404569721453097686010548050360571}{1786733739335630829308913297368119546865050666661771761416660} a^{6} - \frac{5314016815975544837395256847421774508973076294794181036299661}{23227538611363200781015872865785554109245658666603032898416580} a^{5} - \frac{468474739021553452375249104096075577057755770292390463611917}{11613769305681600390507936432892777054622829333301516449208290} a^{4} + \frac{863845252873597042406268187577076920459531092151781198581366}{1935628217613600065084656072148796175770471555550252741534715} a^{3} + \frac{505451981884303523574900275932093854588272797703686352185037}{2322753861136320078101587286578555410924565866660303289841658} a^{2} - \frac{130835283826625399337178532152012952682018947763996708247834}{1935628217613600065084656072148796175770471555550252741534715} a - \frac{161522866277546864734925177226915925309087125789321109633101}{595577913111876943102971099122706515621683555553923920472220}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 999004485714000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{16870}) \), \(\Q(\sqrt{1687}) \), 4.4.75577600.2, 4.4.385600.2, \(\Q(\sqrt{10}, \sqrt{1687})\), 8.8.331757139925442560000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.22.97$x^{8} + 4 x^{7} + 6 x^{4} + 4 x^{2} + 10$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7Data not computed
241Data not computed