↑

Properties

Label 16.0.15340069351...2441.1
Degree $16$
Signature $[0, 8]$
Discriminant $31^{12}\cdot 41^{7}$
Root discriminant $66.70$
Ramified primes $31, 41$
Class number $3$ (GRH)
Class group $[3]$ (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![1500625, 185500, -1169950, -182670, 329692, 96639, -11456, -36480, -14239, 8350, 3572, -1132, -356, 99, 7, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 7*x^14 + 99*x^13 - 356*x^12 - 1132*x^11 + 3572*x^10 + 8350*x^9 - 14239*x^8 - 36480*x^7 - 11456*x^6 + 96639*x^5 + 329692*x^4 - 182670*x^3 - 1169950*x^2 + 185500*x + 1500625)
 
gp: K = bnfinit(x^16 - 5*x^15 + 7*x^14 + 99*x^13 - 356*x^12 - 1132*x^11 + 3572*x^10 + 8350*x^9 - 14239*x^8 - 36480*x^7 - 11456*x^6 + 96639*x^5 + 329692*x^4 - 182670*x^3 - 1169950*x^2 + 185500*x + 1500625, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 7 x^{14} + 99 x^{13} - 356 x^{12} - 1132 x^{11} + 3572 x^{10} + 8350 x^{9} - 14239 x^{8} - 36480 x^{7} - 11456 x^{6} + 96639 x^{5} + 329692 x^{4} - 182670 x^{3} - 1169950 x^{2} + 185500 x + 1500625 \)

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:  \(153400693519826106945591092441=31^{12}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.70$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{13} - \frac{2}{25} a^{12} + \frac{11}{25} a^{11} + \frac{12}{25} a^{10} - \frac{2}{5} a^{9} + \frac{8}{25} a^{8} - \frac{4}{25} a^{7} - \frac{7}{25} a^{6} - \frac{6}{25} a^{3} - \frac{4}{25} a^{2} - \frac{1}{5} a$, $\frac{1}{4375} a^{14} + \frac{38}{4375} a^{13} + \frac{31}{4375} a^{12} - \frac{1998}{4375} a^{11} + \frac{414}{875} a^{10} - \frac{281}{625} a^{9} + \frac{541}{4375} a^{8} + \frac{358}{4375} a^{7} - \frac{28}{125} a^{6} + \frac{17}{35} a^{5} - \frac{308}{625} a^{4} - \frac{444}{4375} a^{3} + \frac{352}{875} a^{2} - \frac{46}{175} a - \frac{1}{5}$, $\frac{1}{196871792317934610358046385140726875} a^{15} - \frac{1532865795854700258730793422589}{196871792317934610358046385140726875} a^{14} + \frac{477699005334383630131531827290176}{39374358463586922071609277028145375} a^{13} + \frac{604776565545522065134033165855243}{39374358463586922071609277028145375} a^{12} - \frac{32778780177494015193750235571023134}{196871792317934610358046385140726875} a^{11} + \frac{68608889007483262178788393374956743}{196871792317934610358046385140726875} a^{10} - \frac{3690327408382154466321833841744876}{7874871692717384414321855405629075} a^{9} + \frac{41266090359280060506622741353492676}{196871792317934610358046385140726875} a^{8} + \frac{2844984457038790967824701900681354}{196871792317934610358046385140726875} a^{7} + \frac{8527595902890533755865383400074197}{39374358463586922071609277028145375} a^{6} - \frac{52411057511593153677812817060093406}{196871792317934610358046385140726875} a^{5} + \frac{72546457349489620083848396324317118}{196871792317934610358046385140726875} a^{4} + \frac{1374061657189587323037365745684723}{196871792317934610358046385140726875} a^{3} + \frac{7110383972276929819196052995974076}{39374358463586922071609277028145375} a^{2} - \frac{2274751608642683759687616624465638}{7874871692717384414321855405629075} a + \frac{29778886627766024993626497059257}{224996334077639554694910154446545}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 381072472.875 \) (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

\(\Q(\sqrt{-31}) \), 4.0.39401.1, 8.0.63649990841.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$31$31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$