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Properties

Label 16.8.62125350736...5625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 11^{2}\cdot 101^{6}\cdot 4451^{2}$
Root discriminant $72.79$
Ramified primes $5, 11, 101, 4451$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T707)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1795, 20300, 32825, -307720, 59511, -25123, 63109, -11743, -12728, 5663, 239, 466, -21, -101, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 28*x^14 - 101*x^13 - 21*x^12 + 466*x^11 + 239*x^10 + 5663*x^9 - 12728*x^8 - 11743*x^7 + 63109*x^6 - 25123*x^5 + 59511*x^4 - 307720*x^3 + 32825*x^2 + 20300*x - 1795)
 
gp: K = bnfinit(x^16 - 28*x^14 - 101*x^13 - 21*x^12 + 466*x^11 + 239*x^10 + 5663*x^9 - 12728*x^8 - 11743*x^7 + 63109*x^6 - 25123*x^5 + 59511*x^4 - 307720*x^3 + 32825*x^2 + 20300*x - 1795, 1)
 

Normalized defining polynomial

\( x^{16} - 28 x^{14} - 101 x^{13} - 21 x^{12} + 466 x^{11} + 239 x^{10} + 5663 x^{9} - 12728 x^{8} - 11743 x^{7} + 63109 x^{6} - 25123 x^{5} + 59511 x^{4} - 307720 x^{3} + 32825 x^{2} + 20300 x - 1795 \)

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:  \(621253507360730723174072265625=5^{12}\cdot 11^{2}\cdot 101^{6}\cdot 4451^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.79$
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:  $5, 11, 101, 4451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{9} + \frac{2}{5} a^{4}$, $\frac{1}{167788725170184916529105166593742037560475} a^{15} + \frac{16715260407278272741856001425735826301607}{167788725170184916529105166593742037560475} a^{14} - \frac{2785826733886532573763163527270257654044}{167788725170184916529105166593742037560475} a^{13} + \frac{6630424312345237340481402656092247828231}{167788725170184916529105166593742037560475} a^{12} + \frac{29680220013114842591373304741398851984431}{167788725170184916529105166593742037560475} a^{11} + \frac{76754154031966469309597208053996254435658}{167788725170184916529105166593742037560475} a^{10} - \frac{2482786268982688247691215625867954076803}{6711549006807396661164206663749681502419} a^{9} - \frac{5304145006086590221217547344536993922207}{15253520470016810593555015144885639778225} a^{8} - \frac{81456125290763238231283672848708745964102}{167788725170184916529105166593742037560475} a^{7} - \frac{6521578705820678168477679464721923145417}{15253520470016810593555015144885639778225} a^{6} - \frac{14865767145738042227404859214332944225244}{33557745034036983305821033318748407512095} a^{5} - \frac{25719050518028518892609796651919413367}{102372620604139668413120907012655300525} a^{4} - \frac{10159920239623654805880231160822934094404}{33557745034036983305821033318748407512095} a^{3} - \frac{13266360956815827151825440245729059102772}{33557745034036983305821033318748407512095} a^{2} + \frac{16190195085132044750397413068742388859491}{33557745034036983305821033318748407512095} a - \frac{4181875685051152827200769725122594383453}{33557745034036983305821033318748407512095}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1035849593.37 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T707):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1, 8.4.312156975625.1, 8.4.788196363453125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
4451Data not computed