↑

Properties

Label 16.0.20396283125...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 109^{14}$
Root discriminant $247.94$
Ramified primes $5, 109$
Class number $535824$ (GRH)
Class group $[2, 366, 732]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9814341056, 12151056320, 13905624448, 5468731520, 1907077408, 369834880, 68749501, 16307885, 1684350, 894085, 57631, -2040, 4163, -45, 58, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 58*x^14 - 45*x^13 + 4163*x^12 - 2040*x^11 + 57631*x^10 + 894085*x^9 + 1684350*x^8 + 16307885*x^7 + 68749501*x^6 + 369834880*x^5 + 1907077408*x^4 + 5468731520*x^3 + 13905624448*x^2 + 12151056320*x + 9814341056)
 
gp: K = bnfinit(x^16 - 5*x^15 + 58*x^14 - 45*x^13 + 4163*x^12 - 2040*x^11 + 57631*x^10 + 894085*x^9 + 1684350*x^8 + 16307885*x^7 + 68749501*x^6 + 369834880*x^5 + 1907077408*x^4 + 5468731520*x^3 + 13905624448*x^2 + 12151056320*x + 9814341056, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 58 x^{14} - 45 x^{13} + 4163 x^{12} - 2040 x^{11} + 57631 x^{10} + 894085 x^{9} + 1684350 x^{8} + 16307885 x^{7} + 68749501 x^{6} + 369834880 x^{5} + 1907077408 x^{4} + 5468731520 x^{3} + 13905624448 x^{2} + 12151056320 x + 9814341056 \)

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:  \(203962831252212947906041415777587890625=5^{14}\cdot 109^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $247.94$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{20} a^{11} - \frac{1}{10} a^{9} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{1}{20} a^{5} - \frac{1}{5} a^{4} + \frac{3}{10} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{80} a^{12} - \frac{1}{40} a^{10} + \frac{1}{16} a^{9} + \frac{1}{10} a^{8} + \frac{3}{20} a^{7} + \frac{11}{80} a^{6} - \frac{1}{20} a^{5} - \frac{1}{8} a^{4} - \frac{9}{80} a^{3} - \frac{1}{4} a^{2} + \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{160} a^{13} - \frac{1}{80} a^{11} - \frac{3}{32} a^{10} + \frac{1}{20} a^{9} + \frac{3}{40} a^{8} + \frac{11}{160} a^{7} + \frac{9}{40} a^{6} + \frac{3}{16} a^{5} + \frac{11}{160} a^{4} + \frac{1}{8} a^{3} - \frac{1}{20} a^{2} - \frac{1}{4} a$, $\frac{1}{23200} a^{14} - \frac{1}{1450} a^{13} - \frac{1}{464} a^{12} + \frac{409}{23200} a^{11} - \frac{297}{2900} a^{10} - \frac{137}{1160} a^{9} + \frac{3}{160} a^{8} - \frac{147}{1160} a^{7} + \frac{79}{464} a^{6} + \frac{199}{4640} a^{5} + \frac{89}{5800} a^{4} - \frac{1277}{2900} a^{3} + \frac{101}{290} a^{2} + \frac{309}{1450} a + \frac{297}{725}$, $\frac{1}{225666443161685260745490619040195360510393644090094753537600} a^{15} + \frac{211092462764004965975487897452627416405100659195251593}{13274496656569721220322977590599727088846684946476161972800} a^{14} - \frac{202465297478927435945246342993248914956636106889740475661}{112833221580842630372745309520097680255196822045047376768800} a^{13} - \frac{1225148713211848032075835985497812066929942064215760936181}{225666443161685260745490619040195360510393644090094753537600} a^{12} + \frac{2127718501357054223259992202935112872423467625880150124897}{225666443161685260745490619040195360510393644090094753537600} a^{11} + \frac{3833007783159821331515002075563787332046695619473568924467}{56416610790421315186372654760048840127598411022523688384400} a^{10} + \frac{712847416677808521605739717876456988832229899404455853011}{9026657726467410429819624761607814420415745763603790141504} a^{9} + \frac{4286493049672474305546738849946204561922457914694585273979}{45133288632337052149098123808039072102078728818018950707520} a^{8} + \frac{892414216971783100129841631287579847847273503019810008077}{4513328863233705214909812380803907210207872881801895070752} a^{7} + \frac{5517326296828292277339701222857080959490944944375079484281}{45133288632337052149098123808039072102078728818018950707520} a^{6} + \frac{13887285538684330670164361760886860794634572284677895219591}{225666443161685260745490619040195360510393644090094753537600} a^{5} + \frac{664186783288073942593160933649844306291293508561526501417}{3318624164142430305080744397649931772211671236619040493200} a^{4} - \frac{1123835336254639075771958419003416406808157461035999265209}{3318624164142430305080744397649931772211671236619040493200} a^{3} + \frac{2920389489954067054918215275852407053411025356252455775403}{28208305395210657593186327380024420063799205511261844192200} a^{2} + \frac{46591797859214766831259183706495590806788893811201094063}{190596658075747686440448157973137973404048685886904352650} a - \frac{52372667425634945649145217151635727709927012026094222837}{542467411446358799868967834231238847380753952139650849850}$

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

Class group and class number

$C_{2}\times C_{366}\times C_{732}$, which has order $535824$ (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:  \( 61163737.6555 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{545}) \), \(\Q(\sqrt{109}) \), 4.4.161878625.1, 4.4.161878625.2, \(\Q(\sqrt{5}, \sqrt{109})\), 8.0.14281555631380390625.2 x2, 8.0.14281555631380390625.1 x2, 8.8.26204689231890625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$109$109.8.7.1$x^{8} - 109$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
109.8.7.1$x^{8} - 109$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$