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Properties

Label 16.0.68681261305...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 29^{4}\cdot 509^{2}$
Root discriminant $41.19$
Ramified primes $2, 5, 11, 29, 509$
Class number $48$ (GRH)
Class group $[2, 24]$ (GRH)
Galois group 16T919

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40229, -134890, 170781, -130456, 64476, 14764, -25976, 15076, -575, -958, 990, -336, 222, -54, 17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 17*x^14 - 54*x^13 + 222*x^12 - 336*x^11 + 990*x^10 - 958*x^9 - 575*x^8 + 15076*x^7 - 25976*x^6 + 14764*x^5 + 64476*x^4 - 130456*x^3 + 170781*x^2 - 134890*x + 40229)
 
gp: K = bnfinit(x^16 - 2*x^15 + 17*x^14 - 54*x^13 + 222*x^12 - 336*x^11 + 990*x^10 - 958*x^9 - 575*x^8 + 15076*x^7 - 25976*x^6 + 14764*x^5 + 64476*x^4 - 130456*x^3 + 170781*x^2 - 134890*x + 40229, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 17 x^{14} - 54 x^{13} + 222 x^{12} - 336 x^{11} + 990 x^{10} - 958 x^{9} - 575 x^{8} + 15076 x^{7} - 25976 x^{6} + 14764 x^{5} + 64476 x^{4} - 130456 x^{3} + 170781 x^{2} - 134890 x + 40229 \)

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:  \(68681261305082905600000000=2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 29^{4}\cdot 509^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.19$
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:  $2, 5, 11, 29, 509$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{30} a^{12} + \frac{2}{5} a^{11} - \frac{1}{15} a^{10} - \frac{1}{3} a^{9} - \frac{13}{30} a^{8} + \frac{4}{15} a^{7} - \frac{1}{6} a^{6} - \frac{2}{5} a^{5} + \frac{13}{30} a^{4} + \frac{2}{15} a^{3} - \frac{4}{15} a^{2} + \frac{7}{15} a - \frac{1}{30}$, $\frac{1}{30} a^{13} + \frac{2}{15} a^{11} + \frac{7}{15} a^{10} - \frac{13}{30} a^{9} + \frac{7}{15} a^{8} - \frac{11}{30} a^{7} - \frac{2}{5} a^{6} + \frac{7}{30} a^{5} - \frac{1}{15} a^{4} + \frac{2}{15} a^{3} - \frac{1}{3} a^{2} + \frac{11}{30} a + \frac{2}{5}$, $\frac{1}{30} a^{14} - \frac{2}{15} a^{11} - \frac{1}{6} a^{10} - \frac{1}{5} a^{9} + \frac{11}{30} a^{8} - \frac{7}{15} a^{7} - \frac{1}{10} a^{6} - \frac{7}{15} a^{5} + \frac{2}{5} a^{4} + \frac{2}{15} a^{3} + \frac{13}{30} a^{2} - \frac{7}{15} a + \frac{2}{15}$, $\frac{1}{26975025751921174798205586173790} a^{15} - \frac{12513207075114879111409738037}{2697502575192117479820558617379} a^{14} - \frac{343661408555338687992774264113}{26975025751921174798205586173790} a^{13} + \frac{135044330832283626189565069151}{13487512875960587399102793086895} a^{12} + \frac{2167316924464024264769453112785}{5395005150384234959641117234758} a^{11} + \frac{37654476002280667461063398156}{385357510741731068545794088197} a^{10} - \frac{1344357929879527302235886600125}{2697502575192117479820558617379} a^{9} - \frac{1725296034555532881591173641639}{4495837625320195799700931028965} a^{8} + \frac{312468431813828580196559465443}{4495837625320195799700931028965} a^{7} + \frac{166104710200542593666696209667}{4495837625320195799700931028965} a^{6} + \frac{12082626309764760532839307580519}{26975025751921174798205586173790} a^{5} + \frac{2570948947568982119158387239659}{13487512875960587399102793086895} a^{4} + \frac{1862114448032637672774062459749}{5395005150384234959641117234758} a^{3} + \frac{88490770845801264341511242108}{4495837625320195799700931028965} a^{2} + \frac{2240531014587479575580732794603}{5395005150384234959641117234758} a + \frac{960106810411759631763082465867}{1926787553708655342728970440985}$

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

Class group and class number

$C_{2}\times C_{24}$, which has order $48$ (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:  \( 29340.4724529 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T919:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 65 conjugacy class representatives for t16n919 are not computed
Character table for t16n919 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, 4.4.725.1, 4.4.127600.1, 8.0.285772960000.1, 8.0.285772960000.2, 8.8.16281760000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
509Data not computed