↑

Properties

Label 16.8.46706312975...8064.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{62}\cdot 17^{6}\cdot 127^{6}$
Root discriminant $261.12$
Ramified primes $2, 17, 127$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1392

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11378288956, -1566486560, -1755236576, -451978368, -434348320, 24897568, 4685344, -2300608, 3455620, -108272, 53536, 7200, -3408, -64, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 - 64*x^13 - 3408*x^12 + 7200*x^11 + 53536*x^10 - 108272*x^9 + 3455620*x^8 - 2300608*x^7 + 4685344*x^6 + 24897568*x^5 - 434348320*x^4 - 451978368*x^3 - 1755236576*x^2 - 1566486560*x + 11378288956)
 
gp: K = bnfinit(x^16 - 48*x^14 - 64*x^13 - 3408*x^12 + 7200*x^11 + 53536*x^10 - 108272*x^9 + 3455620*x^8 - 2300608*x^7 + 4685344*x^6 + 24897568*x^5 - 434348320*x^4 - 451978368*x^3 - 1755236576*x^2 - 1566486560*x + 11378288956, 1)
 

Normalized defining polynomial

\( x^{16} - 48 x^{14} - 64 x^{13} - 3408 x^{12} + 7200 x^{11} + 53536 x^{10} - 108272 x^{9} + 3455620 x^{8} - 2300608 x^{7} + 4685344 x^{6} + 24897568 x^{5} - 434348320 x^{4} - 451978368 x^{3} - 1755236576 x^{2} - 1566486560 x + 11378288956 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(467063129754478147962797661051173208064=2^{62}\cdot 17^{6}\cdot 127^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $261.12$
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:  $2, 17, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{233172} a^{14} - \frac{299}{116586} a^{13} - \frac{431}{25908} a^{12} - \frac{2038}{19431} a^{11} - \frac{2947}{77724} a^{10} - \frac{10}{153} a^{9} - \frac{10946}{58293} a^{8} + \frac{545}{19431} a^{7} - \frac{5782}{19431} a^{6} - \frac{25552}{58293} a^{5} + \frac{46039}{116586} a^{4} + \frac{1886}{19431} a^{3} + \frac{407}{38862} a^{2} - \frac{761}{19431} a - \frac{1154}{58293}$, $\frac{1}{15601094825693534741050225301428281880181960446866443122030142268} a^{15} + \frac{444870154904907166570196971301362155299737346793947674873}{213713627749226503302057880841483313427150143107759494822330716} a^{14} - \frac{127634651348041524112917334712667964608603048258348731510785220}{3900273706423383685262556325357070470045490111716610780507535567} a^{13} + \frac{43529405804231189698766576073413734416519145225684384675674332}{1300091235474461228420852108452356823348496703905536926835845189} a^{12} + \frac{42472249775475750220595549635873246230055762872750747994659811}{1733454980632614971227802811269809097797995605207382569114460252} a^{11} - \frac{22325250211842749655686456949051193411705550502368584521378845}{288909163438769161871300468544968182966332600867897094852410042} a^{10} - \frac{431840034276611160693528244372825831213084204663056209832797633}{15601094825693534741050225301428281880181960446866443122030142268} a^{9} + \frac{1789850131352902053350960627057862631760019331267469744645019439}{7800547412846767370525112650714140940090980223433221561015071134} a^{8} - \frac{44972523681936849096147441224577422333388887656668639352477569}{96303054479589720623766822848322727655444200289299031617470014} a^{7} - \frac{1895142996201955629350575520978849797886673547347442036705807949}{7800547412846767370525112650714140940090980223433221561015071134} a^{6} - \frac{13788226705101835372089931750881387887574821940740291204399135}{152951910055818968049512012759100802746881965165357285510099434} a^{5} + \frac{1112154448605954117826577670260965307331485642986994862309423217}{3900273706423383685262556325357070470045490111716610780507535567} a^{4} - \frac{318942311646297244965073117824816054099380590692590800772190413}{2600182470948922456841704216904713646696993407811073853671690378} a^{3} - \frac{69501106859086081439706652406799172919975593285319072450135237}{433363745158153742806950702817452274449498901301845642278615063} a^{2} - \frac{567929917769598235573755437949856138236935485868581858412013589}{7800547412846767370525112650714140940090980223433221561015071134} a + \frac{5428835565542061459570242044843668953938840027956894068360162}{53428406937306625825514470210370828356787535776939873705582679}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 25357801738500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4421632.2, \(\Q(\zeta_{16})^+\), 4.4.2210816.1, 8.8.312813272694784.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$