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Properties

Label 16.0.61068621502...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{6}\cdot 41^{6}\cdot 97^{4}$
Root discriminant $149.32$
Ramified primes $5, 29, 41, 97$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1134395231, 275105310, 2040819419, 844850241, 655817757, 249045053, 66008293, 14768610, 563022, -846849, -224383, 621, 6821, 237, -108, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 108*x^14 + 237*x^13 + 6821*x^12 + 621*x^11 - 224383*x^10 - 846849*x^9 + 563022*x^8 + 14768610*x^7 + 66008293*x^6 + 249045053*x^5 + 655817757*x^4 + 844850241*x^3 + 2040819419*x^2 + 275105310*x + 1134395231)
 
gp: K = bnfinit(x^16 - x^15 - 108*x^14 + 237*x^13 + 6821*x^12 + 621*x^11 - 224383*x^10 - 846849*x^9 + 563022*x^8 + 14768610*x^7 + 66008293*x^6 + 249045053*x^5 + 655817757*x^4 + 844850241*x^3 + 2040819419*x^2 + 275105310*x + 1134395231, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 108 x^{14} + 237 x^{13} + 6821 x^{12} + 621 x^{11} - 224383 x^{10} - 846849 x^{9} + 563022 x^{8} + 14768610 x^{7} + 66008293 x^{6} + 249045053 x^{5} + 655817757 x^{4} + 844850241 x^{3} + 2040819419 x^{2} + 275105310 x + 1134395231 \)

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:  \(61068621502532689401365830322265625=5^{12}\cdot 29^{6}\cdot 41^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.32$
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:  $5, 29, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{12} + \frac{2}{5} a^{11} - \frac{7}{15} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{15} a^{3} - \frac{7}{15} a^{2} - \frac{4}{15}$, $\frac{1}{45} a^{14} + \frac{2}{45} a^{12} + \frac{1}{9} a^{11} + \frac{8}{45} a^{10} + \frac{1}{5} a^{9} + \frac{1}{15} a^{8} + \frac{2}{5} a^{7} - \frac{1}{15} a^{6} - \frac{1}{3} a^{5} - \frac{14}{45} a^{4} - \frac{4}{15} a^{3} + \frac{11}{45} a^{2} + \frac{14}{45} a - \frac{7}{45}$, $\frac{1}{29143069212086660458132806817394974219981322206182524396557589143335} a^{15} + \frac{982199390643082875894183301683530516889331415237368160179494116}{5828613842417332091626561363478994843996264441236504879311517828667} a^{14} + \frac{331591365167204066044070483177841597594851699545907123229381092249}{29143069212086660458132806817394974219981322206182524396557589143335} a^{13} + \frac{1158318443297097816570690790156375424872081047665764930800028033918}{29143069212086660458132806817394974219981322206182524396557589143335} a^{12} - \frac{235914126663977348982229134131784012149979459039083333575806983146}{5828613842417332091626561363478994843996264441236504879311517828667} a^{11} - \frac{2114509014761258842493248769901839774349510319721062219386257362523}{5828613842417332091626561363478994843996264441236504879311517828667} a^{10} + \frac{4732367720794980534339583556489697509116822140711731508198466807}{9714356404028886819377602272464991406660440735394174798852529714445} a^{9} + \frac{2226211995711017586503952412908121487888189298260679805935653312131}{9714356404028886819377602272464991406660440735394174798852529714445} a^{8} + \frac{3574580097185379119278094224582315051857076642179475370804762701184}{9714356404028886819377602272464991406660440735394174798852529714445} a^{7} - \frac{369441985457212514196153723050775882071027974234710479559553423439}{1079372933780987424375289141384999045184493415043797199872503301605} a^{6} + \frac{3386698572953957697685057554372153092675629847870937178288123426312}{29143069212086660458132806817394974219981322206182524396557589143335} a^{5} + \frac{6317688082589996821773598491626129006784921645891849805877859542119}{29143069212086660458132806817394974219981322206182524396557589143335} a^{4} + \frac{13520448637630253643672274760223870133427985349458890406802240440878}{29143069212086660458132806817394974219981322206182524396557589143335} a^{3} + \frac{50003667830154777438716148634180229710664068258795786160236533724}{200986684221287313504364184947551546344698773835741547562466132023} a^{2} + \frac{11073491902056834953776541581276080431253164879969533925741445601243}{29143069212086660458132806817394974219981322206182524396557589143335} a + \frac{7979177494113519521430145554616175485079888368004266500959434242737}{29143069212086660458132806817394974219981322206182524396557589143335}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 2977910941.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 4.0.29725.1, 4.0.3625.1, 8.0.22089390625.8

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$