↑

Properties

Label 16.8.70704738646...3209.1
Degree $16$
Signature $[8, 4]$
Discriminant $67^{8}\cdot 89^{15}$
Root discriminant $550.29$
Ramified primes $67, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13764977908, -24358063202, 5555106669, 2483202618, 173251248, 180770578, -15297636, -29821162, -6083799, 274386, 125213, -3594, 5002, 574, -150, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 150*x^14 + 574*x^13 + 5002*x^12 - 3594*x^11 + 125213*x^10 + 274386*x^9 - 6083799*x^8 - 29821162*x^7 - 15297636*x^6 + 180770578*x^5 + 173251248*x^4 + 2483202618*x^3 + 5555106669*x^2 - 24358063202*x - 13764977908)
 
gp: K = bnfinit(x^16 - 6*x^15 - 150*x^14 + 574*x^13 + 5002*x^12 - 3594*x^11 + 125213*x^10 + 274386*x^9 - 6083799*x^8 - 29821162*x^7 - 15297636*x^6 + 180770578*x^5 + 173251248*x^4 + 2483202618*x^3 + 5555106669*x^2 - 24358063202*x - 13764977908, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 150 x^{14} + 574 x^{13} + 5002 x^{12} - 3594 x^{11} + 125213 x^{10} + 274386 x^{9} - 6083799 x^{8} - 29821162 x^{7} - 15297636 x^{6} + 180770578 x^{5} + 173251248 x^{4} + 2483202618 x^{3} + 5555106669 x^{2} - 24358063202 x - 13764977908 \)

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:  \(70704738646532877581735658786300135551283209=67^{8}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $550.29$
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:  $67, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{7}{16} a^{3} + \frac{7}{16} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{10} - \frac{1}{32} a^{9} - \frac{3}{64} a^{8} - \frac{3}{16} a^{7} - \frac{5}{32} a^{6} + \frac{3}{16} a^{5} + \frac{9}{64} a^{4} - \frac{1}{4} a^{3} - \frac{17}{64} a^{2} + \frac{1}{32} a + \frac{5}{16}$, $\frac{1}{128} a^{11} - \frac{1}{128} a^{10} + \frac{3}{128} a^{9} - \frac{7}{128} a^{8} + \frac{5}{64} a^{7} - \frac{15}{64} a^{6} - \frac{27}{128} a^{5} - \frac{23}{128} a^{4} + \frac{23}{128} a^{3} - \frac{39}{128} a^{2} - \frac{29}{64} a - \frac{3}{32}$, $\frac{1}{512} a^{12} - \frac{1}{256} a^{11} + \frac{7}{256} a^{9} + \frac{13}{512} a^{8} + \frac{1}{64} a^{7} - \frac{21}{512} a^{6} + \frac{29}{128} a^{5} + \frac{53}{256} a^{4} - \frac{71}{256} a^{3} - \frac{223}{512} a^{2} - \frac{29}{256} a - \frac{33}{128}$, $\frac{1}{11264} a^{13} + \frac{9}{11264} a^{12} - \frac{19}{5632} a^{11} - \frac{41}{5632} a^{10} + \frac{279}{11264} a^{9} - \frac{361}{11264} a^{8} + \frac{2787}{11264} a^{7} + \frac{973}{11264} a^{6} - \frac{1365}{5632} a^{5} - \frac{7}{44} a^{4} - \frac{4649}{11264} a^{3} - \frac{943}{11264} a^{2} + \frac{287}{5632} a - \frac{97}{256}$, $\frac{1}{12525568} a^{14} - \frac{223}{6262784} a^{13} + \frac{135}{12525568} a^{12} - \frac{239}{782848} a^{11} + \frac{101}{12525568} a^{10} + \frac{84759}{6262784} a^{9} - \frac{316329}{6262784} a^{8} - \frac{19623}{142336} a^{7} + \frac{109213}{1138688} a^{6} + \frac{71939}{6262784} a^{5} - \frac{117539}{1138688} a^{4} + \frac{644003}{1565696} a^{3} - \frac{97437}{12525568} a^{2} - \frac{572059}{6262784} a + \frac{37619}{284672}$, $\frac{1}{57528868835173995574960911791029260781723929411141632} a^{15} + \frac{567418716046754684285583928262259505535713345}{57528868835173995574960911791029260781723929411141632} a^{14} - \frac{852029016014088943813540548456603124514506464379}{57528868835173995574960911791029260781723929411141632} a^{13} + \frac{23926668305670123357881248155162486616308925244169}{57528868835173995574960911791029260781723929411141632} a^{12} + \frac{73205392701261966293310212146804968591878314009173}{57528868835173995574960911791029260781723929411141632} a^{11} - \frac{1649194036731473345058130061471338803069695875205}{5229897166833999597723719253729932798338539037376512} a^{10} + \frac{10035162471964684052473005630675625227240440689439}{898888575549593680858764246734832199714436397049088} a^{9} + \frac{991944040045650627739980208772898285884603977258741}{28764434417586997787480455895514630390861964705570816} a^{8} - \frac{890486219869184587330190394476603419396991987780843}{5229897166833999597723719253729932798338539037376512} a^{7} + \frac{7076620204086092420705555607950866933577193605361479}{57528868835173995574960911791029260781723929411141632} a^{6} + \frac{2128188907565218314571621873261681256554186447092985}{57528868835173995574960911791029260781723929411141632} a^{5} + \frac{11328908002017259465409701248652067594132415342611737}{57528868835173995574960911791029260781723929411141632} a^{4} - \frac{884654503039363539492073122131813350500078321113973}{57528868835173995574960911791029260781723929411141632} a^{3} + \frac{22097473629232950948205979946299883006572847511381735}{57528868835173995574960911791029260781723929411141632} a^{2} + \frac{9437875005566028357048095306962841243845075946424573}{28764434417586997787480455895514630390861964705570816} a + \frac{371474926186233214145211736024389591057589317877517}{1307474291708499899430929813432483199584634759344128}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 127446757825000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.891310981471327238009.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89Data not computed