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Properties

Label 16.0.37092500770...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 3^{4}\cdot 5^{8}\cdot 31^{2}\cdot 89^{4}\cdot 33641^{2}$
Root discriminant $222.88$
Ramified primes $2, 3, 5, 31, 89, 33641$
Class number $59550464$ (GRH)
Class group $[2, 2, 2, 2, 3721904]$ (GRH)
Galois group 16T1605

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38179500401221, -17935139332092, 18193806946372, -4967931915742, 1877339814623, -270067793486, 78490251388, -6755823818, 1728619198, -92906620, 21884676, -732770, 159917, -3170, 624, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 624*x^14 - 3170*x^13 + 159917*x^12 - 732770*x^11 + 21884676*x^10 - 92906620*x^9 + 1728619198*x^8 - 6755823818*x^7 + 78490251388*x^6 - 270067793486*x^5 + 1877339814623*x^4 - 4967931915742*x^3 + 18193806946372*x^2 - 17935139332092*x + 38179500401221)
 
gp: K = bnfinit(x^16 - 6*x^15 + 624*x^14 - 3170*x^13 + 159917*x^12 - 732770*x^11 + 21884676*x^10 - 92906620*x^9 + 1728619198*x^8 - 6755823818*x^7 + 78490251388*x^6 - 270067793486*x^5 + 1877339814623*x^4 - 4967931915742*x^3 + 18193806946372*x^2 - 17935139332092*x + 38179500401221, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 624 x^{14} - 3170 x^{13} + 159917 x^{12} - 732770 x^{11} + 21884676 x^{10} - 92906620 x^{9} + 1728619198 x^{8} - 6755823818 x^{7} + 78490251388 x^{6} - 270067793486 x^{5} + 1877339814623 x^{4} - 4967931915742 x^{3} + 18193806946372 x^{2} - 17935139332092 x + 38179500401221 \)

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:  \(37092500770633898047098480230400000000=2^{34}\cdot 3^{4}\cdot 5^{8}\cdot 31^{2}\cdot 89^{4}\cdot 33641^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $222.88$
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:  $2, 3, 5, 31, 89, 33641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{221752859498451338901788725658027736296161292388441305466105797316123514094225097895785228922} a^{15} - \frac{11386514042430977445616754713581898354372645628536788380738202556893339458078962119828143293}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{14} - \frac{27516549325399326951437542221842848922063861400855040078005451203089994419627908353918259738}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{13} + \frac{869676933529007755591107385262323018221220184276031278395783448181783471492750432764164111}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{12} + \frac{493429971455920262809655338773537646632664598627545330801351797775428868086702177313712171}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{11} - \frac{51901577835157480050311132799409655007142197811168781469245591032712640240985480929535405942}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{10} + \frac{34323286214683710345652364497681575165326601801174154879326717960061104981776120417558842248}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{9} + \frac{37085233717020143944066909909407034854628974522010213127923756753192953296011609943811970276}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{8} - \frac{32599934690816841729541939309112030439789935171624958177499123051076422246422951435295021204}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{7} - \frac{16357363331491555066572716112021473176983377550632344001369946452286458439420650507220420991}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{6} - \frac{17319074393218309554227058420081027516277637917734640079537486010581452803813269978242393583}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{5} + \frac{54533657605372103157287098711865556814483216281858108356788634787650948174797277017598718740}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{4} + \frac{31213285838188841342471755204025971636023345807122362831282840772688022768196761223827435145}{221752859498451338901788725658027736296161292388441305466105797316123514094225097895785228922} a^{3} + \frac{34672720821933793908965118628869461393011329814787224348781041840927998087921963262889246334}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a^{2} - \frac{47083692531472970544333926823731098513471696891429665154871081568901622082695276071682544387}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461} a + \frac{14920371310315617565796199415797051180199963139623501168210768793881588812319935832972962491}{110876429749225669450894362829013868148080646194220652733052898658061757047112548947892614461}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{3721904}$, which has order $59550464$ (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:  \( 21348.0660855 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.5069440000.1

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 R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.22.49$x^{8} + 4 x^{6} + 6 x^{4} + 52$$4$$2$$22$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
2.8.12.24$x^{8} + 4 x^{6} + 28 x^{4} + 80$$4$$2$$12$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
33641Data not computed