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Properties

Label 16.4.90819108489...7168.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{56}\cdot 3^{8}\cdot 577^{3}$
Root discriminant $64.55$
Ramified primes $2, 3, 577$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1379

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1981832, 2109376, 2988832, 3657696, 3150192, 1334528, 835200, 66016, 84364, -30832, 4048, -4624, 388, -176, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 - 176*x^13 + 388*x^12 - 4624*x^11 + 4048*x^10 - 30832*x^9 + 84364*x^8 + 66016*x^7 + 835200*x^6 + 1334528*x^5 + 3150192*x^4 + 3657696*x^3 + 2988832*x^2 + 2109376*x - 1981832)
 
gp: K = bnfinit(x^16 + 40*x^14 - 176*x^13 + 388*x^12 - 4624*x^11 + 4048*x^10 - 30832*x^9 + 84364*x^8 + 66016*x^7 + 835200*x^6 + 1334528*x^5 + 3150192*x^4 + 3657696*x^3 + 2988832*x^2 + 2109376*x - 1981832, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} - 176 x^{13} + 388 x^{12} - 4624 x^{11} + 4048 x^{10} - 30832 x^{9} + 84364 x^{8} + 66016 x^{7} + 835200 x^{6} + 1334528 x^{5} + 3150192 x^{4} + 3657696 x^{3} + 2988832 x^{2} + 2109376 x - 1981832 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(90819108489560418567132807168=2^{56}\cdot 3^{8}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.55$
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:  $2, 3, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{4} a^{14}$, $\frac{1}{152657778190598927950014063627263057424764181412} a^{15} - \frac{1368134741470818880415879251655287843646178463}{38164444547649731987503515906815764356191045353} a^{14} + \frac{4560444272790951481545477793711502572035976925}{38164444547649731987503515906815764356191045353} a^{13} + \frac{3915622425847160860702538899148653801118857261}{38164444547649731987503515906815764356191045353} a^{12} - \frac{784621206799038461865448400299128483497003400}{38164444547649731987503515906815764356191045353} a^{11} - \frac{18751903057360481379395083813562592890030692339}{76328889095299463975007031813631528712382090706} a^{10} - \frac{1988382647529510705686569915109247695540376331}{38164444547649731987503515906815764356191045353} a^{9} + \frac{5444428636882813739582379730286411271149684607}{76328889095299463975007031813631528712382090706} a^{8} - \frac{1695180341124555689329581357704526731878624810}{38164444547649731987503515906815764356191045353} a^{7} + \frac{5670183603275858720882243743595195829738753718}{38164444547649731987503515906815764356191045353} a^{6} + \frac{3931910127193828323347476698991636293059697874}{38164444547649731987503515906815764356191045353} a^{5} - \frac{238533097609732163035953519399286059092342179}{537527387995066647711317125448109357129451343} a^{4} + \frac{15590310241592938817729568082787402660052637477}{38164444547649731987503515906815764356191045353} a^{3} + \frac{15354325660432162124204785607568412744476644809}{38164444547649731987503515906815764356191045353} a^{2} + \frac{14508097321470182990284890012891326773471136649}{38164444547649731987503515906815764356191045353} a + \frac{11668378572743616189018475530706677218619612449}{38164444547649731987503515906815764356191045353}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 50142296.8093 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1379:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1379 are not computed
Character table for t16n1379 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{16})^+\), 4.4.18432.1, \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{48})^+\)

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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed
577Data not computed