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Properties

Label 16.8.14557329728...7184.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{14}\cdot 367^{4}$
Root discriminant $32.37$
Ramified primes $2, 3, 367$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1049

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -16, 74, -24, -406, 190, 466, -678, -488, 456, 148, -178, 17, 48, -16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 16*x^14 + 48*x^13 + 17*x^12 - 178*x^11 + 148*x^10 + 456*x^9 - 488*x^8 - 678*x^7 + 466*x^6 + 190*x^5 - 406*x^4 - 24*x^3 + 74*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 16*x^14 + 48*x^13 + 17*x^12 - 178*x^11 + 148*x^10 + 456*x^9 - 488*x^8 - 678*x^7 + 466*x^6 + 190*x^5 - 406*x^4 - 24*x^3 + 74*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 16 x^{14} + 48 x^{13} + 17 x^{12} - 178 x^{11} + 148 x^{10} + 456 x^{9} - 488 x^{8} - 678 x^{7} + 466 x^{6} + 190 x^{5} - 406 x^{4} - 24 x^{3} + 74 x^{2} - 16 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1455732972800792995037184=2^{24}\cdot 3^{14}\cdot 367^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.37$
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:  $2, 3, 367$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{5}{13} a^{11} - \frac{1}{13} a^{10} - \frac{1}{13} a^{9} - \frac{6}{13} a^{8} - \frac{3}{13} a^{7} - \frac{5}{13} a^{6} + \frac{4}{13} a^{5} + \frac{5}{13} a^{4} + \frac{4}{13} a^{3} - \frac{6}{13} a^{2} + \frac{3}{13} a + \frac{1}{13}$, $\frac{1}{169} a^{13} + \frac{3}{13} a^{11} - \frac{74}{169} a^{10} - \frac{14}{169} a^{9} - \frac{77}{169} a^{8} - \frac{3}{169} a^{7} + \frac{29}{169} a^{6} - \frac{67}{169} a^{5} + \frac{31}{169} a^{4} - \frac{84}{169} a^{2} - \frac{14}{169} a + \frac{60}{169}$, $\frac{1}{2197} a^{14} - \frac{5}{2197} a^{13} + \frac{3}{169} a^{12} - \frac{269}{2197} a^{11} + \frac{187}{2197} a^{10} - \frac{514}{2197} a^{9} + \frac{382}{2197} a^{8} + \frac{720}{2197} a^{7} - \frac{550}{2197} a^{6} - \frac{986}{2197} a^{5} - \frac{1000}{2197} a^{4} - \frac{1098}{2197} a^{3} + \frac{237}{2197} a^{2} - \frac{55}{169} a + \frac{714}{2197}$, $\frac{1}{251936581} a^{15} - \frac{36085}{251936581} a^{14} + \frac{631162}{251936581} a^{13} - \frac{7499670}{251936581} a^{12} - \frac{85573789}{251936581} a^{11} - \frac{84533104}{251936581} a^{10} + \frac{55037334}{251936581} a^{9} + \frac{8537821}{251936581} a^{8} + \frac{110854390}{251936581} a^{7} + \frac{61993473}{251936581} a^{6} - \frac{31176050}{251936581} a^{5} + \frac{100134127}{251936581} a^{4} - \frac{23279639}{251936581} a^{3} + \frac{21782473}{251936581} a^{2} - \frac{60063381}{251936581} a + \frac{89127108}{251936581}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 969062.196948 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1049:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 31 conjugacy class representatives for t16n1049
Character table for t16n1049 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.9909.1, 8.8.25136199936.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed
367Data not computed