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Properties

Label 16.0.80994658951...0704.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{50}\cdot 449^{2}\cdot 1889^{2}$
Root discriminant $48.06$
Ramified primes $2, 449, 1889$
Class number $256$ (GRH)
Class group $[2, 2, 64]$ (GRH)
Galois group 16T1720

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23428, 19312, 28712, 42912, 47084, 48816, 34928, 20688, 11460, 3672, 1852, 192, 182, -8, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 8*x^13 + 182*x^12 + 192*x^11 + 1852*x^10 + 3672*x^9 + 11460*x^8 + 20688*x^7 + 34928*x^6 + 48816*x^5 + 47084*x^4 + 42912*x^3 + 28712*x^2 + 19312*x + 23428)
 
gp: K = bnfinit(x^16 + 16*x^14 - 8*x^13 + 182*x^12 + 192*x^11 + 1852*x^10 + 3672*x^9 + 11460*x^8 + 20688*x^7 + 34928*x^6 + 48816*x^5 + 47084*x^4 + 42912*x^3 + 28712*x^2 + 19312*x + 23428, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 8 x^{13} + 182 x^{12} + 192 x^{11} + 1852 x^{10} + 3672 x^{9} + 11460 x^{8} + 20688 x^{7} + 34928 x^{6} + 48816 x^{5} + 47084 x^{4} + 42912 x^{3} + 28712 x^{2} + 19312 x + 23428 \)

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:  \(809946589519572593702600704=2^{50}\cdot 449^{2}\cdot 1889^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.06$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{35898246998332795273311841404068} a^{15} - \frac{3358017101711630057619227325519}{35898246998332795273311841404068} a^{14} + \frac{104083846606910802162248156539}{8974561749583198818327960351017} a^{13} - \frac{2481250056829211100184231443675}{35898246998332795273311841404068} a^{12} + \frac{599447013483004638173913241607}{17949123499166397636655920702034} a^{11} - \frac{1028593959938346461634632018257}{17949123499166397636655920702034} a^{10} + \frac{1205558223077600754070551886962}{8974561749583198818327960351017} a^{9} - \frac{897924505792033346035908204573}{17949123499166397636655920702034} a^{8} - \frac{6624438639012952989997477481937}{17949123499166397636655920702034} a^{7} - \frac{6035321011858185193206754883535}{17949123499166397636655920702034} a^{6} + \frac{4176677293416146449456750456857}{8974561749583198818327960351017} a^{5} - \frac{1275189978588267583378341222747}{17949123499166397636655920702034} a^{4} + \frac{1109674396063242453593742975356}{8974561749583198818327960351017} a^{3} - \frac{4066952447193011560140037716585}{8974561749583198818327960351017} a^{2} + \frac{4210586169709066399822048971446}{8974561749583198818327960351017} a - \frac{4430488314240012271823838232606}{8974561749583198818327960351017}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{64}$, which has order $256$ (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:  \( 56271.9156358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1720:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 116 conjugacy class representatives for t16n1720 are not computed
Character table for t16n1720 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.26.8$x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
449Data not computed
1889Data not computed