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Properties

Label 16.0.10271172687...5625.9
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{10}$
Root discriminant $27.43$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19321, 23352, -3020, -6636, 11890, 6008, -7911, -4767, 2426, 1622, -498, -388, 78, 53, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 6*x^14 + 53*x^13 + 78*x^12 - 388*x^11 - 498*x^10 + 1622*x^9 + 2426*x^8 - 4767*x^7 - 7911*x^6 + 6008*x^5 + 11890*x^4 - 6636*x^3 - 3020*x^2 + 23352*x + 19321)
 
gp: K = bnfinit(x^16 - 4*x^15 - 6*x^14 + 53*x^13 + 78*x^12 - 388*x^11 - 498*x^10 + 1622*x^9 + 2426*x^8 - 4767*x^7 - 7911*x^6 + 6008*x^5 + 11890*x^4 - 6636*x^3 - 3020*x^2 + 23352*x + 19321, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 6 x^{14} + 53 x^{13} + 78 x^{12} - 388 x^{11} - 498 x^{10} + 1622 x^{9} + 2426 x^{8} - 4767 x^{7} - 7911 x^{6} + 6008 x^{5} + 11890 x^{4} - 6636 x^{3} - 3020 x^{2} + 23352 x + 19321 \)

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:  \(102711726879931884765625=5^{12}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.43$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{12} - \frac{4}{11} a^{11} - \frac{1}{11} a^{10} - \frac{3}{11} a^{9} - \frac{3}{11} a^{8} + \frac{3}{11} a^{7} - \frac{1}{11} a^{6} + \frac{4}{11} a^{5} - \frac{2}{11} a^{3} + \frac{3}{11} a^{2} + \frac{3}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{12} - \frac{2}{11} a^{11} + \frac{5}{11} a^{10} - \frac{1}{11} a^{9} + \frac{5}{11} a^{8} - \frac{3}{11} a^{7} + \frac{1}{11} a^{6} + \frac{1}{11} a^{5} - \frac{2}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} - \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{75415381048716485148527598509} a^{15} + \frac{297854559770116930450328101}{75415381048716485148527598509} a^{14} - \frac{3166771074276493160890279278}{75415381048716485148527598509} a^{13} + \frac{6438216592316277523536826778}{75415381048716485148527598509} a^{12} + \frac{15073930875851525940448864014}{75415381048716485148527598509} a^{11} + \frac{20038890342192383306609151489}{75415381048716485148527598509} a^{10} - \frac{21369557694256471520825833303}{75415381048716485148527598509} a^{9} - \frac{35490183821977104770000314987}{75415381048716485148527598509} a^{8} - \frac{31350997173067169886069506297}{75415381048716485148527598509} a^{7} - \frac{23468184420458031797543050195}{75415381048716485148527598509} a^{6} - \frac{21168152417215760918213286219}{75415381048716485148527598509} a^{5} - \frac{36381195492199495720944676704}{75415381048716485148527598509} a^{4} + \frac{3012704765659324506480493237}{75415381048716485148527598509} a^{3} - \frac{4280652097086112489340334976}{75415381048716485148527598509} a^{2} - \frac{30823838027190193692738996293}{75415381048716485148527598509} a + \frac{142598304025579942858155132}{542556698192204929126097831}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{10377150903472986182246187}{75415381048716485148527598509} a^{15} - \frac{49557449689653310769574142}{75415381048716485148527598509} a^{14} - \frac{5614250530788930548703825}{75415381048716485148527598509} a^{13} + \frac{475924305803915935111489191}{75415381048716485148527598509} a^{12} + \frac{445927076449646889256881386}{75415381048716485148527598509} a^{11} - \frac{3810197425717723932442788950}{75415381048716485148527598509} a^{10} - \frac{1029033852011724607484056173}{75415381048716485148527598509} a^{9} + \frac{14065080381515717233366852833}{75415381048716485148527598509} a^{8} + \frac{9116028669549235926501187831}{75415381048716485148527598509} a^{7} - \frac{45622754061557073512886241756}{75415381048716485148527598509} a^{6} - \frac{16698164700205457635702477741}{75415381048716485148527598509} a^{5} + \frac{45678153914191243370318110264}{75415381048716485148527598509} a^{4} + \frac{12627897768269906925647917522}{75415381048716485148527598509} a^{3} - \frac{59619656982785363859278467515}{75415381048716485148527598509} a^{2} + \frac{65956213202332044314217595492}{75415381048716485148527598509} a + \frac{455871323310180544658229269}{542556698192204929126097831} \) (order $10$)
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:  \( 86779.6039367 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, \(\Q(\zeta_{5})\), 4.0.3625.1, 8.4.320486703125.1, 8.4.12819468125.1, 8.0.13140625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$