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Properties

Label 16.8.23563276747...0553.1
Degree $16$
Signature $[8, 4]$
Discriminant $61^{2}\cdot 97^{15}$
Root discriminant $121.83$
Ramified primes $61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4070861, -12535249, 23108071, 5401798, -15046663, 742099, 2009890, -188533, 186871, 8184, -31773, 2279, -223, -193, 52, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 52*x^14 - 193*x^13 - 223*x^12 + 2279*x^11 - 31773*x^10 + 8184*x^9 + 186871*x^8 - 188533*x^7 + 2009890*x^6 + 742099*x^5 - 15046663*x^4 + 5401798*x^3 + 23108071*x^2 - 12535249*x - 4070861)
 
gp: K = bnfinit(x^16 - x^15 + 52*x^14 - 193*x^13 - 223*x^12 + 2279*x^11 - 31773*x^10 + 8184*x^9 + 186871*x^8 - 188533*x^7 + 2009890*x^6 + 742099*x^5 - 15046663*x^4 + 5401798*x^3 + 23108071*x^2 - 12535249*x - 4070861, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 52 x^{14} - 193 x^{13} - 223 x^{12} + 2279 x^{11} - 31773 x^{10} + 8184 x^{9} + 186871 x^{8} - 188533 x^{7} + 2009890 x^{6} + 742099 x^{5} - 15046663 x^{4} + 5401798 x^{3} + 23108071 x^{2} - 12535249 x - 4070861 \)

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:  \(2356327674777993305467029827040553=61^{2}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.83$
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:  $61, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{35679708576067477719526718755090596764065681617481862696353} a^{15} + \frac{17486049884488387796336773959586970398923544352087184041519}{35679708576067477719526718755090596764065681617481862696353} a^{14} - \frac{5338916158337090784891904009214390182248565771852358546}{184868956352681231707392325155909827793086433251201361121} a^{13} + \frac{30482258771335944853176412312717098334863882676221584907}{184868956352681231707392325155909827793086433251201361121} a^{12} + \frac{15556200836322027080367406054770628676478118186464642887483}{35679708576067477719526718755090596764065681617481862696353} a^{11} + \frac{14629447716323392128712767962782677586523290369275455061122}{35679708576067477719526718755090596764065681617481862696353} a^{10} - \frac{525987509701933618116191834886093456664591993439099183111}{35679708576067477719526718755090596764065681617481862696353} a^{9} - \frac{13409831778347950346621400445706587065483286649273776115886}{35679708576067477719526718755090596764065681617481862696353} a^{8} + \frac{3614889898490208618244954935244620321402696192225598146641}{35679708576067477719526718755090596764065681617481862696353} a^{7} + \frac{11969873641910836290210052114112328148683904091995846618941}{35679708576067477719526718755090596764065681617481862696353} a^{6} - \frac{15377267181874637599863396149716639654310541305907813253350}{35679708576067477719526718755090596764065681617481862696353} a^{5} + \frac{9267924309719009050170364598605777832538262317157451945567}{35679708576067477719526718755090596764065681617481862696353} a^{4} + \frac{12167868937783021633302858463296248963823844648172968130432}{35679708576067477719526718755090596764065681617481862696353} a^{3} - \frac{1936685810587635485437774932422840537500614597317378602538}{35679708576067477719526718755090596764065681617481862696353} a^{2} + \frac{7665613840230576516548625245264150007106924706684483576488}{35679708576067477719526718755090596764065681617481862696353} a + \frac{6829921904620007028015520636081683637158905251367348377241}{35679708576067477719526718755090596764065681617481862696353}$

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:  \( 19425520160.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
97Data not computed