↑

Properties

Label 16.0.85304566600...5184.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 449^{2}\cdot 1889^{5}$
Root discriminant $152.47$
Ramified primes $2, 449, 1889$
Class number $12354304$ (GRH)
Class group $[2, 2, 4, 772144]$ (GRH)
Galois group 16T1186

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![112207690208, -15531166208, 56568573472, -6154880192, 11255370396, -906010208, 1132616672, -60883952, 62512596, -1978792, 1970244, -39808, 34241, -572, 298, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 298*x^14 - 572*x^13 + 34241*x^12 - 39808*x^11 + 1970244*x^10 - 1978792*x^9 + 62512596*x^8 - 60883952*x^7 + 1132616672*x^6 - 906010208*x^5 + 11255370396*x^4 - 6154880192*x^3 + 56568573472*x^2 - 15531166208*x + 112207690208)
 
gp: K = bnfinit(x^16 - 4*x^15 + 298*x^14 - 572*x^13 + 34241*x^12 - 39808*x^11 + 1970244*x^10 - 1978792*x^9 + 62512596*x^8 - 60883952*x^7 + 1132616672*x^6 - 906010208*x^5 + 11255370396*x^4 - 6154880192*x^3 + 56568573472*x^2 - 15531166208*x + 112207690208, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 298 x^{14} - 572 x^{13} + 34241 x^{12} - 39808 x^{11} + 1970244 x^{10} - 1978792 x^{9} + 62512596 x^{8} - 60883952 x^{7} + 1132616672 x^{6} - 906010208 x^{5} + 11255370396 x^{4} - 6154880192 x^{3} + 56568573472 x^{2} - 15531166208 x + 112207690208 \)

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:  \(85304566600455667746642216756445184=2^{44}\cdot 449^{2}\cdot 1889^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.47$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{248} a^{14} + \frac{7}{124} a^{13} - \frac{1}{62} a^{12} - \frac{15}{124} a^{11} + \frac{19}{248} a^{10} - \frac{15}{124} a^{9} - \frac{11}{124} a^{8} + \frac{21}{124} a^{7} - \frac{17}{124} a^{6} - \frac{3}{31} a^{5} + \frac{13}{31} a^{4} - \frac{19}{62} a^{3} - \frac{9}{31} a^{2} + \frac{8}{31} a + \frac{2}{31}$, $\frac{1}{9869703778122425138798847401542386355014713225431135520250013241744} a^{15} + \frac{43528179220209587640216033654169029762403989642428167760294223}{31233239804184889679743187979564513781692130460225112405854472284} a^{14} + \frac{25686634893920810661341123632182377631518568797983689725850850843}{2467425944530606284699711850385596588753678306357783880062503310436} a^{13} - \frac{75651395133669723356003460056816392079540062077921101872936835495}{1233712972265303142349855925192798294376839153178891940031251655218} a^{12} + \frac{134704330008889332027589655494429162725444345264490132122696071325}{9869703778122425138798847401542386355014713225431135520250013241744} a^{11} + \frac{289916081494871257043437042723772050819114100163265692271158573531}{2467425944530606284699711850385596588753678306357783880062503310436} a^{10} + \frac{327087447169785916125822097944880910835547446204528230221739393063}{4934851889061212569399423700771193177507356612715567760125006620872} a^{9} + \frac{28144036488033282006889053276247801328502662521287739470753557287}{1233712972265303142349855925192798294376839153178891940031251655218} a^{8} + \frac{24067236099909110354690941446963548648684773828906715182291519852}{616856486132651571174927962596399147188419576589445970015625827609} a^{7} - \frac{82568262901801105185301292630241771664898243285999757404508574780}{616856486132651571174927962596399147188419576589445970015625827609} a^{6} + \frac{43905436938626345685603228506037397945687499223215948526258523617}{2467425944530606284699711850385596588753678306357783880062503310436} a^{5} + \frac{376795848892699719426310142793803946993941281971909338476044399287}{1233712972265303142349855925192798294376839153178891940031251655218} a^{4} - \frac{143737529667550298364351433421890075507793871759869476561007484955}{2467425944530606284699711850385596588753678306357783880062503310436} a^{3} + \frac{145036166125313574477320831299414833200448129907340724777374632904}{616856486132651571174927962596399147188419576589445970015625827609} a^{2} + \frac{229889515567109012222118172528492527662759520995426705258454456794}{616856486132651571174927962596399147188419576589445970015625827609} a - \frac{221742704326477365164308510508429554423798430584829796493878487665}{616856486132651571174927962596399147188419576589445970015625827609}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{772144}$, which has order $12354304$ (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:  \( 56271.9156358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1186:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 46 conjugacy class representatives for t16n1186
Character table for t16n1186 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
449Data not computed
1889Data not computed