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Properties

Label 16.16.3373848744...7936.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{56}\cdot 193^{2}\cdot 257^{10}$
Root discriminant $700.66$
Ramified primes $2, 193, 257$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group 16T1435

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9841036804, 0, -127780508968, 0, 74304624364, 0, -13117641856, 0, 768110204, 0, -18900004, 0, 199222, 0, -808, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 808*x^14 + 199222*x^12 - 18900004*x^10 + 768110204*x^8 - 13117641856*x^6 + 74304624364*x^4 - 127780508968*x^2 + 9841036804)
 
gp: K = bnfinit(x^16 - 808*x^14 + 199222*x^12 - 18900004*x^10 + 768110204*x^8 - 13117641856*x^6 + 74304624364*x^4 - 127780508968*x^2 + 9841036804, 1)
 

Normalized defining polynomial

\( x^{16} - 808 x^{14} + 199222 x^{12} - 18900004 x^{10} + 768110204 x^{8} - 13117641856 x^{6} + 74304624364 x^{4} - 127780508968 x^{2} + 9841036804 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3373848744484085385907843975364516141319847936=2^{56}\cdot 193^{2}\cdot 257^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $700.66$
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:  $2, 193, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{1028} a^{12} + \frac{55}{257} a^{10} - \frac{105}{514} a^{8} - \frac{56}{257} a^{6} + \frac{213}{514} a^{4}$, $\frac{1}{1028} a^{13} + \frac{55}{257} a^{11} - \frac{105}{514} a^{9} - \frac{56}{257} a^{7} + \frac{213}{514} a^{5}$, $\frac{1}{26324077903014818106390606696360044407484} a^{14} - \frac{8996004527394458459432679141449266469}{26324077903014818106390606696360044407484} a^{12} - \frac{518935282292611923307244209055458060357}{6581019475753704526597651674090011101871} a^{10} - \frac{628750328827865416235030902751489807401}{13162038951507409053195303348180022203742} a^{8} - \frac{4980313714749264103315951344127908097177}{13162038951507409053195303348180022203742} a^{6} + \frac{3324203602543441033826271321803441589987}{13162038951507409053195303348180022203742} a^{4} + \frac{9560733775107600813374958076521460297}{25607079672193402827228216630700432303} a^{2} + \frac{21237646203078572504230530493568081}{132679169285976180451959671661660271}$, $\frac{1}{26324077903014818106390606696360044407484} a^{15} - \frac{8996004527394458459432679141449266469}{26324077903014818106390606696360044407484} a^{13} - \frac{518935282292611923307244209055458060357}{6581019475753704526597651674090011101871} a^{11} - \frac{628750328827865416235030902751489807401}{13162038951507409053195303348180022203742} a^{9} - \frac{4980313714749264103315951344127908097177}{13162038951507409053195303348180022203742} a^{7} + \frac{3324203602543441033826271321803441589987}{13162038951507409053195303348180022203742} a^{5} + \frac{9560733775107600813374958076521460297}{25607079672193402827228216630700432303} a^{3} + \frac{21237646203078572504230530493568081}{132679169285976180451959671661660271} a$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  $15$
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:  \( 71568111279500000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1435:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 56 conjugacy class representatives for t16n1435 are not computed
Character table for t16n1435 is not computed

Intermediate fields

\(\Q(\sqrt{514}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{257}) \), \(\Q(\sqrt{2}, \sqrt{257})\), 8.8.2287190881599488.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.25.2$x^{8} + 10 x^{4} + 20 x^{2} + 2$$8$$1$$25$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
2.8.31.12$x^{8} + 8 x^{6} + 12 x^{4} + 2$$8$$1$$31$$C_8:C_2$$[3, 4, 5]^{2}$
193Data not computed
257Data not computed