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Properties

Label 16.0.18843687673...1961.2
Degree $16$
Signature $[0, 8]$
Discriminant $41^{15}\cdot 59^{4}$
Root discriminant $90.09$
Ramified primes $41, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29720377, -5993851, 15334849, -3327225, 3489306, -1023430, 486411, -168474, 46307, -11277, 2904, -198, 141, -48, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 13*x^14 - 48*x^13 + 141*x^12 - 198*x^11 + 2904*x^10 - 11277*x^9 + 46307*x^8 - 168474*x^7 + 486411*x^6 - 1023430*x^5 + 3489306*x^4 - 3327225*x^3 + 15334849*x^2 - 5993851*x + 29720377)
 
gp: K = bnfinit(x^16 - 5*x^15 + 13*x^14 - 48*x^13 + 141*x^12 - 198*x^11 + 2904*x^10 - 11277*x^9 + 46307*x^8 - 168474*x^7 + 486411*x^6 - 1023430*x^5 + 3489306*x^4 - 3327225*x^3 + 15334849*x^2 - 5993851*x + 29720377, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 13 x^{14} - 48 x^{13} + 141 x^{12} - 198 x^{11} + 2904 x^{10} - 11277 x^{9} + 46307 x^{8} - 168474 x^{7} + 486411 x^{6} - 1023430 x^{5} + 3489306 x^{4} - 3327225 x^{3} + 15334849 x^{2} - 5993851 x + 29720377 \)

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:  \(18843687673244176811423009101961=41^{15}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.09$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{59} a^{14} - \frac{24}{59} a^{13} + \frac{10}{59} a^{12} - \frac{19}{59} a^{11} - \frac{17}{59} a^{10} - \frac{4}{59} a^{9} - \frac{14}{59} a^{8} + \frac{29}{59} a^{7} + \frac{26}{59} a^{6} - \frac{28}{59} a^{5} - \frac{26}{59} a^{3} + \frac{9}{59} a^{2} - \frac{16}{59} a - \frac{10}{59}$, $\frac{1}{44236589400716312547015942850610142493622391379} a^{15} - \frac{356511771731366186207286449232687106919755914}{44236589400716312547015942850610142493622391379} a^{14} + \frac{6440424807115535230829862930054666357869754887}{44236589400716312547015942850610142493622391379} a^{13} + \frac{14190423093430338270259753357316185195993663239}{44236589400716312547015942850610142493622391379} a^{12} + \frac{9947782773255443978319915908580503456712779455}{44236589400716312547015942850610142493622391379} a^{11} + \frac{15765215743832710088018746176765737080427790335}{44236589400716312547015942850610142493622391379} a^{10} + \frac{13359593397254503620878499858290051380594313072}{44236589400716312547015942850610142493622391379} a^{9} + \frac{12264790846208764033444651895207538511637578935}{44236589400716312547015942850610142493622391379} a^{8} + \frac{232404645653726182519669728935053179203397559}{44236589400716312547015942850610142493622391379} a^{7} - \frac{10945253150458338396757448389195833824066837044}{44236589400716312547015942850610142493622391379} a^{6} + \frac{21137540011911532906433227800953822184648683948}{44236589400716312547015942850610142493622391379} a^{5} - \frac{21974091213724257149648993002977864085802849796}{44236589400716312547015942850610142493622391379} a^{4} - \frac{2374290432057742334119443956488059470276310613}{44236589400716312547015942850610142493622391379} a^{3} - \frac{8919188204425088126084073493440741549351510865}{44236589400716312547015942850610142493622391379} a^{2} + \frac{20528907068988339053378371172034328302938683770}{44236589400716312547015942850610142493622391379} a + \frac{5208530258623036376105668739665938547808012385}{44236589400716312547015942850610142493622391379}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 574982744.485 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.0.194754273881.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ R

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
41Data not computed
$59$59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$