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Properties

Label 16.16.3373848744...7936.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{56}\cdot 193^{2}\cdot 257^{10}$
Root discriminant $700.66$
Ramified primes $2, 193, 257$
Class number $24$ (GRH)
Class group $[2, 2, 6]$ (GRH)
Galois group 16T1435

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![157456588864, 0, -651038123904, 0, 231712491168, 0, -25675961248, 0, 1204861360, 0, -25038032, 0, 215956, 0, -788, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 788*x^14 + 215956*x^12 - 25038032*x^10 + 1204861360*x^8 - 25675961248*x^6 + 231712491168*x^4 - 651038123904*x^2 + 157456588864)
 
gp: K = bnfinit(x^16 - 788*x^14 + 215956*x^12 - 25038032*x^10 + 1204861360*x^8 - 25675961248*x^6 + 231712491168*x^4 - 651038123904*x^2 + 157456588864, 1)
 

Normalized defining polynomial

\( x^{16} - 788 x^{14} + 215956 x^{12} - 25038032 x^{10} + 1204861360 x^{8} - 25675961248 x^{6} + 231712491168 x^{4} - 651038123904 x^{2} + 157456588864 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3373848744484085385907843975364516141319847936=2^{56}\cdot 193^{2}\cdot 257^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $700.66$
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:  $2, 193, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{8} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{144} a^{10} - \frac{1}{18} a^{8} + \frac{1}{36} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{2}{9}$, $\frac{1}{144} a^{11} - \frac{1}{18} a^{9} + \frac{1}{36} a^{7} + \frac{1}{6} a^{5} - \frac{2}{9} a$, $\frac{1}{666144} a^{12} + \frac{317}{166536} a^{10} + \frac{1097}{41634} a^{8} + \frac{253}{41634} a^{6} + \frac{3175}{27756} a^{4} + \frac{16}{81} a^{2} + \frac{31}{81}$, $\frac{1}{666144} a^{13} + \frac{317}{166536} a^{11} + \frac{1097}{41634} a^{9} + \frac{253}{41634} a^{7} + \frac{3175}{27756} a^{5} + \frac{16}{81} a^{3} + \frac{31}{81} a$, $\frac{1}{103982832080751572124971871698095968} a^{14} + \frac{9053182465880538704899058419}{17330472013458595354161978616349328} a^{12} - \frac{3233130602831413957374688297235}{3058318590610340356616819755826352} a^{10} + \frac{247444574830618511195594990554171}{8665236006729297677080989308174664} a^{8} - \frac{954634657001843520232360731155467}{12997854010093946515621483962261996} a^{6} + \frac{20324553345458197127586350072483}{249958730963345125300413153120423} a^{4} + \frac{7341109210968234940103077315123}{25287653716136082715216894868214} a^{2} - \frac{14293644722283091440791875466}{65512056259419903407297655099}$, $\frac{1}{207965664161503144249943743396191936} a^{15} + \frac{9053182465880538704899058419}{34660944026917190708323957232698656} a^{13} + \frac{562662279471053981500621666889}{191144911913146272288551234739147} a^{11} + \frac{53074817206120678117610925226366}{1083154500841162209635123663521833} a^{9} + \frac{2655880345802030511884718147250643}{25995708020187893031242967924523992} a^{7} + \frac{123968683678698102688643751185107}{999834923853380501201652612481692} a^{5} - \frac{1325679411774951604376342529746}{12643826858068041357608447434107} a^{3} - \frac{14425939723299312765651232744}{65512056259419903407297655099} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$ (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:  $15$
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:  \( 118207782376000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1435:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 56 conjugacy class representatives for t16n1435 are not computed
Character table for t16n1435 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{514}) \), \(\Q(\sqrt{257}) \), \(\Q(\sqrt{2}, \sqrt{257})\), 8.8.2287190881599488.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.31.12$x^{8} + 8 x^{6} + 12 x^{4} + 2$$8$$1$$31$$C_8:C_2$$[3, 4, 5]^{2}$
2.8.25.2$x^{8} + 10 x^{4} + 20 x^{2} + 2$$8$$1$$25$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
193Data not computed
257Data not computed