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Properties

Label 16.0.60651338988...0253.2
Degree $16$
Signature $[0, 8]$
Discriminant $37^{3}\cdot 149^{12}$
Root discriminant $83.93$
Ramified primes $37, 149$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4409, -18586, 90463, -130628, 52452, -31491, 93551, -123761, 98643, -47496, 14604, -2066, -167, 124, -18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 18*x^14 + 124*x^13 - 167*x^12 - 2066*x^11 + 14604*x^10 - 47496*x^9 + 98643*x^8 - 123761*x^7 + 93551*x^6 - 31491*x^5 + 52452*x^4 - 130628*x^3 + 90463*x^2 - 18586*x + 4409)
 
gp: K = bnfinit(x^16 - 3*x^15 - 18*x^14 + 124*x^13 - 167*x^12 - 2066*x^11 + 14604*x^10 - 47496*x^9 + 98643*x^8 - 123761*x^7 + 93551*x^6 - 31491*x^5 + 52452*x^4 - 130628*x^3 + 90463*x^2 - 18586*x + 4409, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 18 x^{14} + 124 x^{13} - 167 x^{12} - 2066 x^{11} + 14604 x^{10} - 47496 x^{9} + 98643 x^{8} - 123761 x^{7} + 93551 x^{6} - 31491 x^{5} + 52452 x^{4} - 130628 x^{3} + 90463 x^{2} - 18586 x + 4409 \)

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:  \(6065133898851787352862994830253=37^{3}\cdot 149^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.93$
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:  $37, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{31} a^{12} + \frac{1}{31} a^{11} - \frac{14}{31} a^{10} - \frac{10}{31} a^{9} + \frac{10}{31} a^{8} + \frac{12}{31} a^{7} - \frac{13}{31} a^{6} - \frac{4}{31} a^{5} + \frac{15}{31} a^{4} - \frac{14}{31} a^{3} + \frac{10}{31} a^{2} - \frac{11}{31} a - \frac{14}{31}$, $\frac{1}{31} a^{13} - \frac{15}{31} a^{11} + \frac{4}{31} a^{10} - \frac{11}{31} a^{9} + \frac{2}{31} a^{8} + \frac{6}{31} a^{7} + \frac{9}{31} a^{6} - \frac{12}{31} a^{5} + \frac{2}{31} a^{4} - \frac{7}{31} a^{3} + \frac{10}{31} a^{2} - \frac{3}{31} a + \frac{14}{31}$, $\frac{1}{252743} a^{14} - \frac{1611}{252743} a^{13} - \frac{3831}{252743} a^{12} + \frac{32908}{252743} a^{11} - \frac{69250}{252743} a^{10} + \frac{122223}{252743} a^{9} - \frac{5788}{252743} a^{8} + \frac{59313}{252743} a^{7} + \frac{74653}{252743} a^{6} + \frac{91700}{252743} a^{5} - \frac{114967}{252743} a^{4} + \frac{27666}{252743} a^{3} - \frac{109391}{252743} a^{2} + \frac{41553}{252743} a - \frac{57697}{252743}$, $\frac{1}{44880100659786031605607495812689} a^{15} - \frac{599476231538097840686749}{669852248653522859785186504667} a^{14} - \frac{719636692214420303061156201982}{44880100659786031605607495812689} a^{13} - \frac{363789790688588589583772577531}{44880100659786031605607495812689} a^{12} + \frac{19484408619099428684927817414026}{44880100659786031605607495812689} a^{11} + \frac{1309947420408633540709103912754}{44880100659786031605607495812689} a^{10} + \frac{18012168324254026243385654484671}{44880100659786031605607495812689} a^{9} + \frac{12456390787871719015199337810839}{44880100659786031605607495812689} a^{8} - \frac{15006353789935114709732215757485}{44880100659786031605607495812689} a^{7} - \frac{20456785647061333523886098611732}{44880100659786031605607495812689} a^{6} + \frac{21846247017865368244439358414674}{44880100659786031605607495812689} a^{5} - \frac{416392462222040824962411021322}{2640005921163884212094558577217} a^{4} - \frac{12421692102219116018877830865843}{44880100659786031605607495812689} a^{3} - \frac{21908885427224786801530369708011}{44880100659786031605607495812689} a^{2} - \frac{22019694249821298367225242504278}{44880100659786031605607495812689} a - \frac{1104749155544801855454586667210}{44880100659786031605607495812689}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( 138559515.792 \) (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

\(\Q(\sqrt{149}) \), 4.0.3307949.1, 8.0.404873483704237.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
37Data not computed
149Data not computed