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Properties

Label 16.10.3193845368...4779.1
Degree $16$
Signature $[10, 3]$
Discriminant $-\,41^{15}\cdot 59^{3}$
Root discriminant $69.83$
Ramified primes $41, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10000, 15600, -172416, -261716, 103863, 243758, 30286, -41894, -20367, -9164, -891, 1510, 447, -28, -34, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 34*x^14 - 28*x^13 + 447*x^12 + 1510*x^11 - 891*x^10 - 9164*x^9 - 20367*x^8 - 41894*x^7 + 30286*x^6 + 243758*x^5 + 103863*x^4 - 261716*x^3 - 172416*x^2 + 15600*x + 10000)
 
gp: K = bnfinit(x^16 - 2*x^15 - 34*x^14 - 28*x^13 + 447*x^12 + 1510*x^11 - 891*x^10 - 9164*x^9 - 20367*x^8 - 41894*x^7 + 30286*x^6 + 243758*x^5 + 103863*x^4 - 261716*x^3 - 172416*x^2 + 15600*x + 10000, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 34 x^{14} - 28 x^{13} + 447 x^{12} + 1510 x^{11} - 891 x^{10} - 9164 x^{9} - 20367 x^{8} - 41894 x^{7} + 30286 x^{6} + 243758 x^{5} + 103863 x^{4} - 261716 x^{3} - 172416 x^{2} + 15600 x + 10000 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-319384536834647064600389984779=-\,41^{15}\cdot 59^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{3}{10} a^{6} + \frac{1}{10} a^{5} - \frac{1}{10} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{3}{10} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{2}{5} a^{5} - \frac{3}{10} a^{4} + \frac{2}{5} a^{3} + \frac{1}{10} a$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{9} - \frac{1}{5} a^{8} - \frac{1}{20} a^{7} - \frac{1}{10} a^{6} - \frac{7}{20} a^{5} + \frac{2}{5} a^{4} - \frac{3}{10} a^{3} - \frac{2}{5} a^{2} - \frac{7}{20} a - \frac{1}{2}$, $\frac{1}{80} a^{14} - \frac{1}{40} a^{13} + \frac{1}{40} a^{12} - \frac{1}{20} a^{11} + \frac{3}{16} a^{10} - \frac{1}{8} a^{9} - \frac{3}{16} a^{8} + \frac{1}{20} a^{7} - \frac{11}{80} a^{6} - \frac{3}{8} a^{5} - \frac{7}{40} a^{4} + \frac{3}{40} a^{3} - \frac{5}{16} a^{2} + \frac{1}{20} a - \frac{1}{4}$, $\frac{1}{23933488777308445851935566842584000} a^{15} + \frac{28085665716188124205657297495537}{5983372194327111462983891710646000} a^{14} + \frac{92634404668050756138370527869083}{11966744388654222925967783421292000} a^{13} - \frac{25867361968166471418766230651491}{2991686097163555731491945855323000} a^{12} + \frac{593435532373984798571049440150447}{23933488777308445851935566842584000} a^{11} + \frac{30865021437685099921005998731901}{149584304858177786574597292766150} a^{10} + \frac{1461690346462916229261290208136709}{23933488777308445851935566842584000} a^{9} + \frac{343972801867139153048140899943593}{11966744388654222925967783421292000} a^{8} - \frac{4372384496330959899527710469178467}{23933488777308445851935566842584000} a^{7} - \frac{1323000660786462756985866532890343}{2991686097163555731491945855323000} a^{6} - \frac{2509883827553793043787409512780257}{11966744388654222925967783421292000} a^{5} - \frac{491751988997422403115748781113471}{11966744388654222925967783421292000} a^{4} - \frac{8095527893091323106834800671411037}{23933488777308445851935566842584000} a^{3} - \frac{1725466679658690182037122873847633}{11966744388654222925967783421292000} a^{2} - \frac{1686618850611111919658057752831579}{5983372194327111462983891710646000} a - \frac{33325308928748503526882312298891}{119667443886542229259677834212920}$

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:  $12$
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:  \( 2893408318.83 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.11490502158979.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ $16$ 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
59Data not computed