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Properties

Label 16.0.85974764485...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{8}\cdot 89^{6}\cdot 97^{4}$
Root discriminant $203.42$
Ramified primes $5, 29, 89, 97$
Class number $47220672$ (GRH)
Class group $[2, 2, 2, 5902584]$ (GRH)
Galois group $(C_2^2\times D_4).C_2^3$ (as 16T600)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12702224754731, -7615234442750, 5885733201608, -1958997450216, 797569771891, -166583131684, 46617802187, -6041597096, 1345512595, -97838278, 20443090, -784824, 165956, -2924, 661, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 661*x^14 - 2924*x^13 + 165956*x^12 - 784824*x^11 + 20443090*x^10 - 97838278*x^9 + 1345512595*x^8 - 6041597096*x^7 + 46617802187*x^6 - 166583131684*x^5 + 797569771891*x^4 - 1958997450216*x^3 + 5885733201608*x^2 - 7615234442750*x + 12702224754731)
 
gp: K = bnfinit(x^16 - 4*x^15 + 661*x^14 - 2924*x^13 + 165956*x^12 - 784824*x^11 + 20443090*x^10 - 97838278*x^9 + 1345512595*x^8 - 6041597096*x^7 + 46617802187*x^6 - 166583131684*x^5 + 797569771891*x^4 - 1958997450216*x^3 + 5885733201608*x^2 - 7615234442750*x + 12702224754731, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 661 x^{14} - 2924 x^{13} + 165956 x^{12} - 784824 x^{11} + 20443090 x^{10} - 97838278 x^{9} + 1345512595 x^{8} - 6041597096 x^{7} + 46617802187 x^{6} - 166583131684 x^{5} + 797569771891 x^{4} - 1958997450216 x^{3} + 5885733201608 x^{2} - 7615234442750 x + 12702224754731 \)

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:  \(8597476448565520621040798398594140625=5^{8}\cdot 29^{8}\cdot 89^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.42$
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:  $5, 29, 89, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2180} a^{14} + \frac{181}{1090} a^{13} - \frac{44}{545} a^{12} + \frac{171}{1090} a^{11} + \frac{241}{1090} a^{10} + \frac{35}{218} a^{9} + \frac{88}{545} a^{8} + \frac{66}{545} a^{7} - \frac{909}{2180} a^{6} + \frac{337}{1090} a^{5} + \frac{121}{1090} a^{4} - \frac{197}{545} a^{3} - \frac{5}{436} a^{2} + \frac{403}{1090} a + \frac{519}{2180}$, $\frac{1}{162099142619278434356160290488264458417060648134517733420103356727006657564097320222991080} a^{15} - \frac{22027876786166604767537845712223567092055860878330223733425740057200898094893321568347}{162099142619278434356160290488264458417060648134517733420103356727006657564097320222991080} a^{14} + \frac{2335722450621884421452849750890665515989919196354676940463857172751709021136602330497}{44313598310354957451109975529869999567266442901727100442893208509296516556614904380260} a^{13} + \frac{15373373753053615462379003476581899888428269004678193962746997009170230210631968056107733}{81049571309639217178080145244132229208530324067258866710051678363503328782048660111495540} a^{12} + \frac{19765988696384330767188219072684979020068086801564290619694548166978968949241158729091627}{81049571309639217178080145244132229208530324067258866710051678363503328782048660111495540} a^{11} + \frac{19674191578453760301708413060328276588268388584010455524897046903372148156713424030792221}{81049571309639217178080145244132229208530324067258866710051678363503328782048660111495540} a^{10} - \frac{4882261110954088136717385202028990411159442398255970240939087017853626143286425852306936}{20262392827409804294520036311033057302132581016814716677512919590875832195512165027873885} a^{9} - \frac{12222226981079059877939749420463494391053398979912582186545573636204342782903141298644947}{81049571309639217178080145244132229208530324067258866710051678363503328782048660111495540} a^{8} + \frac{5881423190519790016806964834035739973830031170213331964417169195286764153370821889476653}{32419828523855686871232058097652891683412129626903546684020671345401331512819464044598216} a^{7} + \frac{589027781681492908507415115604872540034107640482136464493303977383247331294784815023153}{32419828523855686871232058097652891683412129626903546684020671345401331512819464044598216} a^{6} - \frac{7792826490205122798133746760887516461932850676558694480405500958202065195866329540995943}{20262392827409804294520036311033057302132581016814716677512919590875832195512165027873885} a^{5} - \frac{11434125318799345784941383581317332295171108634736064057676825780772070222048571919291409}{40524785654819608589040072622066114604265162033629433355025839181751664391024330055747770} a^{4} - \frac{55780825200723417022933823414712267205585194131827819218674600395703068932150231624298093}{162099142619278434356160290488264458417060648134517733420103356727006657564097320222991080} a^{3} + \frac{20618033122702097783012978863974367950250875879454526147546066861752686184263094685984191}{162099142619278434356160290488264458417060648134517733420103356727006657564097320222991080} a^{2} - \frac{4576772196651120225161060864971092097512895641276244686077267103245638598286387097468133}{32419828523855686871232058097652891683412129626903546684020671345401331512819464044598216} a - \frac{978104256694053143192350266947062768572092627636842233029975347648340816093500466475391}{162099142619278434356160290488264458417060648134517733420103356727006657564097320222991080}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{5902584}$, which has order $47220672$ (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:  \( 13998.0176198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times D_4).C_2^3$ (as 16T600):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $(C_2^2\times D_4).C_2^3$
Character table for $(C_2^2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.1, 4.4.2225.1, 4.4.725.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$