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Properties

Label 16.0.26893256677...0000.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 29^{6}\cdot 41^{4}$
Root discriminant $59.82$
Ramified primes $2, 5, 29, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2376465001, 0, 1275176342, 0, 254177286, 0, 25996296, 0, 1843044, 0, 86088, 0, 3069, 0, 64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 64*x^14 + 3069*x^12 + 86088*x^10 + 1843044*x^8 + 25996296*x^6 + 254177286*x^4 + 1275176342*x^2 + 2376465001)
 
gp: K = bnfinit(x^16 + 64*x^14 + 3069*x^12 + 86088*x^10 + 1843044*x^8 + 25996296*x^6 + 254177286*x^4 + 1275176342*x^2 + 2376465001, 1)
 

Normalized defining polynomial

\( x^{16} + 64 x^{14} + 3069 x^{12} + 86088 x^{10} + 1843044 x^{8} + 25996296 x^{6} + 254177286 x^{4} + 1275176342 x^{2} + 2376465001 \)

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:  \(26893256677956496000000000000=2^{16}\cdot 5^{12}\cdot 29^{6}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.82$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{82} a^{10} - \frac{9}{41} a^{8} + \frac{35}{82} a^{6} - \frac{6}{41} a^{4} - \frac{29}{82} a^{2} - \frac{1}{2}$, $\frac{1}{82} a^{11} - \frac{9}{41} a^{9} + \frac{35}{82} a^{7} - \frac{6}{41} a^{5} - \frac{29}{82} a^{3} - \frac{1}{2} a$, $\frac{1}{877482} a^{12} - \frac{125}{97498} a^{10} - \frac{12139}{438741} a^{8} - \frac{2138}{438741} a^{6} + \frac{151097}{877482} a^{4} - \frac{7}{123} a^{2} - \frac{4}{9}$, $\frac{1}{877482} a^{13} - \frac{125}{97498} a^{11} - \frac{12139}{438741} a^{9} - \frac{2138}{438741} a^{7} + \frac{151097}{877482} a^{5} - \frac{7}{123} a^{3} - \frac{4}{9} a$, $\frac{1}{3012988357696166740680282} a^{14} + \frac{838879075864528163}{1506494178848083370340141} a^{12} + \frac{9892654622798631740275}{3012988357696166740680282} a^{10} + \frac{1930609873006798320515}{11544016696153895558162} a^{8} - \frac{328445222417042699156428}{1506494178848083370340141} a^{6} - \frac{566359591043175419579}{73487520919418700992202} a^{4} - \frac{350769832297605002}{753733624478642649} a^{2} - \frac{144277252783631023}{1507467248957285298}$, $\frac{1}{3012988357696166740680282} a^{15} + \frac{838879075864528163}{1506494178848083370340141} a^{13} + \frac{9892654622798631740275}{3012988357696166740680282} a^{11} + \frac{1930609873006798320515}{11544016696153895558162} a^{9} - \frac{328445222417042699156428}{1506494178848083370340141} a^{7} - \frac{566359591043175419579}{73487520919418700992202} a^{5} - \frac{350769832297605002}{753733624478642649} a^{3} - \frac{144277252783631023}{1507467248957285298} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -\frac{79988873607551}{5783087058917786450442} a^{14} - \frac{26007495595040}{33236132522516014083} a^{12} - \frac{105985575216717574}{2891543529458893225221} a^{10} - \frac{2665917183679642775}{2891543529458893225221} a^{8} - \frac{54815578402795488344}{2891543529458893225221} a^{6} - \frac{3613556896269400117}{15672322652893730218} a^{4} - \frac{244844562866604293}{118629860282627058} a^{2} - \frac{5312469003013645}{964470408801846} \) (order $10$)
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:  \( 44471073.6358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.725.1, 4.0.3625.1, 8.0.13140625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed