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Properties

Label 16.8.16564827256...7089.3
Degree $16$
Signature $[8, 4]$
Discriminant $13^{14}\cdot 29^{10}$
Root discriminant $77.39$
Ramified primes $13, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16448, 138288, -732188, 901634, -65817, -352485, -7647, 117118, 7346, -18627, -4367, 1705, 578, -42, -37, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 37*x^14 - 42*x^13 + 578*x^12 + 1705*x^11 - 4367*x^10 - 18627*x^9 + 7346*x^8 + 117118*x^7 - 7647*x^6 - 352485*x^5 - 65817*x^4 + 901634*x^3 - 732188*x^2 + 138288*x + 16448)
 
gp: K = bnfinit(x^16 - x^15 - 37*x^14 - 42*x^13 + 578*x^12 + 1705*x^11 - 4367*x^10 - 18627*x^9 + 7346*x^8 + 117118*x^7 - 7647*x^6 - 352485*x^5 - 65817*x^4 + 901634*x^3 - 732188*x^2 + 138288*x + 16448, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 37 x^{14} - 42 x^{13} + 578 x^{12} + 1705 x^{11} - 4367 x^{10} - 18627 x^{9} + 7346 x^{8} + 117118 x^{7} - 7647 x^{6} - 352485 x^{5} - 65817 x^{4} + 901634 x^{3} - 732188 x^{2} + 138288 x + 16448 \)

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:  \(1656482725689092973620649257089=13^{14}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.39$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{18} a^{12} + \frac{4}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{5}{18} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{18} a^{13} + \frac{1}{9} a^{10} + \frac{1}{6} a^{8} + \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{7}{18} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{108} a^{14} + \frac{1}{108} a^{13} - \frac{1}{108} a^{12} + \frac{2}{27} a^{11} + \frac{1}{54} a^{10} + \frac{37}{108} a^{9} + \frac{23}{108} a^{8} + \frac{29}{108} a^{7} + \frac{8}{27} a^{6} - \frac{1}{54} a^{5} - \frac{11}{108} a^{4} + \frac{1}{12} a^{3} - \frac{41}{108} a^{2} + \frac{10}{27} a + \frac{4}{27}$, $\frac{1}{63260735439337590343802205375726768} a^{15} + \frac{16314519136446152005201932773783}{63260735439337590343802205375726768} a^{14} - \frac{329839784426032493940777724314503}{21086911813112530114600735125242256} a^{13} - \frac{62293578282869865260405481192529}{31630367719668795171901102687863384} a^{12} - \frac{58231383600342807952136485947067}{31630367719668795171901102687863384} a^{11} + \frac{838594675104955071204913645842353}{7028970604370843371533578375080752} a^{10} + \frac{5029312011452244505719852593945795}{21086911813112530114600735125242256} a^{9} + \frac{20736364682306584608697272365286253}{63260735439337590343802205375726768} a^{8} - \frac{3095714282795439858159105720044059}{31630367719668795171901102687863384} a^{7} - \frac{2253843900707109729835856059461343}{10543455906556265057300367562621128} a^{6} - \frac{11304580328242416256835327843409247}{63260735439337590343802205375726768} a^{5} + \frac{14741779975303499108887134589794547}{63260735439337590343802205375726768} a^{4} - \frac{24793411129502456933500988443501241}{63260735439337590343802205375726768} a^{3} + \frac{4440799574250604306678148575381021}{31630367719668795171901102687863384} a^{2} + \frac{5284482353382870673182474290083559}{15815183859834397585950551343931692} a - \frac{560211451785042721878606855127361}{3953795964958599396487637835982923}$

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:  \( 3189409072.55 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{29}) \), 4.4.1847677.1 x2, 4.4.63713.1 x2, \(\Q(\sqrt{13}, \sqrt{29})\), 8.8.3413910296329.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$