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Properties

Label 16.0.74256122186...2833.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{15}\cdot 29^{9}$
Root discriminant $73.61$
Ramified primes $13, 29$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10037, -19659, 15330, 7123, -4459, -28119, -2834, 14196, 4680, 4719, 1872, 806, 715, -66, 68, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 68*x^14 - 66*x^13 + 715*x^12 + 806*x^11 + 1872*x^10 + 4719*x^9 + 4680*x^8 + 14196*x^7 - 2834*x^6 - 28119*x^5 - 4459*x^4 + 7123*x^3 + 15330*x^2 - 19659*x + 10037)
 
gp: K = bnfinit(x^16 - 3*x^15 + 68*x^14 - 66*x^13 + 715*x^12 + 806*x^11 + 1872*x^10 + 4719*x^9 + 4680*x^8 + 14196*x^7 - 2834*x^6 - 28119*x^5 - 4459*x^4 + 7123*x^3 + 15330*x^2 - 19659*x + 10037, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 68 x^{14} - 66 x^{13} + 715 x^{12} + 806 x^{11} + 1872 x^{10} + 4719 x^{9} + 4680 x^{8} + 14196 x^{7} - 2834 x^{6} - 28119 x^{5} - 4459 x^{4} + 7123 x^{3} + 15330 x^{2} - 19659 x + 10037 \)

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:  \(742561221860627884726497942833=13^{15}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.61$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{12} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{5}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{5}{12} a - \frac{5}{12}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{12} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{4}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{12} - \frac{5}{24} a^{11} + \frac{5}{24} a^{10} + \frac{1}{12} a^{9} + \frac{1}{24} a^{8} - \frac{1}{2} a^{7} + \frac{11}{24} a^{6} + \frac{3}{8} a^{5} + \frac{5}{12} a^{4} + \frac{1}{12} a^{3} - \frac{11}{24} a^{2} - \frac{1}{3} a + \frac{5}{24}$, $\frac{1}{3594479582760013210584768133283762328} a^{15} + \frac{8141497372501975830827348236496200}{449309947845001651323096016660470291} a^{14} - \frac{14177461294954774204176307892356687}{1198159860920004403528256044427920776} a^{13} + \frac{8769495832985778430765333526723509}{399386620306668134509418681475973592} a^{12} - \frac{629852457925159434674541953049340829}{3594479582760013210584768133283762328} a^{11} + \frac{25874215593802858248552013698444452}{449309947845001651323096016660470291} a^{10} + \frac{1200821178515433812710894870153470467}{3594479582760013210584768133283762328} a^{9} + \frac{504124793044699346648188916337337625}{1797239791380006605292384066641881164} a^{8} + \frac{815332986248673384731223951511106339}{3594479582760013210584768133283762328} a^{7} - \frac{1439651596505729555715697253592717503}{3594479582760013210584768133283762328} a^{6} + \frac{72675119671598216966410418331998579}{199693310153334067254709340737986796} a^{5} - \frac{73117621448499978677462003192759257}{299539965230001100882064011106980194} a^{4} - \frac{1428751885354033208594341931235231547}{3594479582760013210584768133283762328} a^{3} - \frac{685868851276402635299432614537286495}{1797239791380006605292384066641881164} a^{2} - \frac{412446926702269199979596838777386357}{3594479582760013210584768133283762328} a + \frac{819760184574704897243422824807355987}{1797239791380006605292384066641881164}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 71251312.8914 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.1530373581113.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $16$

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
13Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$