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Properties

Label 16.0.15321543254...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{8}\cdot 941^{4}$
Root discriminant $66.69$
Ramified primes $5, 29, 941$
Class number $12992$ (GRH)
Class group $[2, 2, 3248]$ (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T394)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50639329, -28773927, 45696860, -17389496, 17725464, -5423962, 4121484, -830772, 591665, -84419, 52307, -5607, 2798, -205, 82, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 82*x^14 - 205*x^13 + 2798*x^12 - 5607*x^11 + 52307*x^10 - 84419*x^9 + 591665*x^8 - 830772*x^7 + 4121484*x^6 - 5423962*x^5 + 17725464*x^4 - 17389496*x^3 + 45696860*x^2 - 28773927*x + 50639329)
 
gp: K = bnfinit(x^16 - 3*x^15 + 82*x^14 - 205*x^13 + 2798*x^12 - 5607*x^11 + 52307*x^10 - 84419*x^9 + 591665*x^8 - 830772*x^7 + 4121484*x^6 - 5423962*x^5 + 17725464*x^4 - 17389496*x^3 + 45696860*x^2 - 28773927*x + 50639329, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 82 x^{14} - 205 x^{13} + 2798 x^{12} - 5607 x^{11} + 52307 x^{10} - 84419 x^{9} + 591665 x^{8} - 830772 x^{7} + 4121484 x^{6} - 5423962 x^{5} + 17725464 x^{4} - 17389496 x^{3} + 45696860 x^{2} - 28773927 x + 50639329 \)

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:  \(153215432545895383062469140625=5^{8}\cdot 29^{8}\cdot 941^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.69$
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:  $5, 29, 941$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{38581} a^{14} - \frac{307}{38581} a^{13} + \frac{7259}{38581} a^{12} + \frac{12548}{38581} a^{11} - \frac{2955}{38581} a^{10} - \frac{399}{941} a^{9} - \frac{9901}{38581} a^{8} + \frac{12189}{38581} a^{7} + \frac{4992}{38581} a^{6} + \frac{8220}{38581} a^{5} - \frac{1183}{38581} a^{4} - \frac{17184}{38581} a^{3} + \frac{437}{941} a^{2} - \frac{7660}{38581} a + \frac{11917}{38581}$, $\frac{1}{73567512947823854313720789009611119068186529140149} a^{15} - \frac{546677731060474158224356319849201770797616223}{73567512947823854313720789009611119068186529140149} a^{14} - \frac{13256564120331689477390788772249667501894000371556}{73567512947823854313720789009611119068186529140149} a^{13} + \frac{5574716345218088348315400986743052396219759542397}{73567512947823854313720789009611119068186529140149} a^{12} - \frac{10060312703501404690830752418430379648532430933561}{73567512947823854313720789009611119068186529140149} a^{11} + \frac{26427674622495539632836863906382355958026518978865}{73567512947823854313720789009611119068186529140149} a^{10} - \frac{15832261967043255179095710751664033809508904723285}{73567512947823854313720789009611119068186529140149} a^{9} + \frac{8071538530607395717234420997521869578635849229284}{73567512947823854313720789009611119068186529140149} a^{8} + \frac{21052587488571560029764786295742397933494633221447}{73567512947823854313720789009611119068186529140149} a^{7} + \frac{23986574374017649294662507630436305022480349508571}{73567512947823854313720789009611119068186529140149} a^{6} - \frac{19078153278065054393027566144814246488885595044480}{73567512947823854313720789009611119068186529140149} a^{5} - \frac{26768112665412491801080273136158720968354595884894}{73567512947823854313720789009611119068186529140149} a^{4} - \frac{7749994170907580745251508885034350239429409083548}{73567512947823854313720789009611119068186529140149} a^{3} - \frac{16980915760531391707560989132708571299689931975385}{73567512947823854313720789009611119068186529140149} a^{2} + \frac{4548116420259728922379276558022108719230570051200}{73567512947823854313720789009611119068186529140149} a - \frac{1532337840145822645370889617773766315434401062817}{5659039457524911870286214539200855312937425318473}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3248}$, which has order $12992$ (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:  \( 3793.72993285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T394):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 17 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
$29$29.4.2.1$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}$
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}$
941Data not computed