↑

Properties

Label 16.0.26996664364...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 89^{7}$
Root discriminant $29.14$
Ramified primes $5, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![401, 2218, 5024, 6438, 5443, 3843, 4151, 3529, 1638, 313, 229, 254, 123, 6, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 9*x^14 + 6*x^13 + 123*x^12 + 254*x^11 + 229*x^10 + 313*x^9 + 1638*x^8 + 3529*x^7 + 4151*x^6 + 3843*x^5 + 5443*x^4 + 6438*x^3 + 5024*x^2 + 2218*x + 401)
 
gp: K = bnfinit(x^16 - x^15 - 9*x^14 + 6*x^13 + 123*x^12 + 254*x^11 + 229*x^10 + 313*x^9 + 1638*x^8 + 3529*x^7 + 4151*x^6 + 3843*x^5 + 5443*x^4 + 6438*x^3 + 5024*x^2 + 2218*x + 401, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 9 x^{14} + 6 x^{13} + 123 x^{12} + 254 x^{11} + 229 x^{10} + 313 x^{9} + 1638 x^{8} + 3529 x^{7} + 4151 x^{6} + 3843 x^{5} + 5443 x^{4} + 6438 x^{3} + 5024 x^{2} + 2218 x + 401 \)

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:  \(269966643649468994140625=5^{14}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.14$
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:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11301104424949398320667686251} a^{15} + \frac{4073891999579854998936209589}{11301104424949398320667686251} a^{14} + \frac{5064127686195204642057681002}{11301104424949398320667686251} a^{13} + \frac{2991925437843022491156283101}{11301104424949398320667686251} a^{12} - \frac{5593842418944482160976202487}{11301104424949398320667686251} a^{11} - \frac{2769573940659014921120373472}{11301104424949398320667686251} a^{10} + \frac{2109524697297418649848131091}{11301104424949398320667686251} a^{9} + \frac{1449495814145229152041690096}{11301104424949398320667686251} a^{8} + \frac{97778596562816747072007501}{11301104424949398320667686251} a^{7} - \frac{3266015738527536775031609597}{11301104424949398320667686251} a^{6} - \frac{3867515227387425605103118403}{11301104424949398320667686251} a^{5} + \frac{5245064494096825278108874878}{11301104424949398320667686251} a^{4} - \frac{3027709670782940840616496808}{11301104424949398320667686251} a^{3} + \frac{4458564796944522901582497514}{11301104424949398320667686251} a^{2} - \frac{1999350639465310638384453498}{11301104424949398320667686251} a + \frac{3990813608154083629449817695}{11301104424949398320667686251}$

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:  $7$
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:  \( 24104.2464739 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.11015140625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$89$89.8.0.1$x^{8} - x + 62$$1$$8$$0$$C_8$$[\ ]^{8}$
89.8.7.8$x^{8} + 194643$$8$$1$$7$$C_8$$[\ ]_{8}$