↑

Properties

Label 16.0.41476437673...5625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{4}\cdot 29^{6}$
Root discriminant $22.44$
Ramified primes $5, 13, 29$
Class number $6$
Class group $[6]$
Galois group $C_2^4.C_2^3.C_2$ (as 16T500)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1831, 1108, 1642, -6182, 5215, -3845, 3856, -3184, 2381, -1691, 1106, -606, 283, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 283*x^12 - 606*x^11 + 1106*x^10 - 1691*x^9 + 2381*x^8 - 3184*x^7 + 3856*x^6 - 3845*x^5 + 5215*x^4 - 6182*x^3 + 1642*x^2 + 1108*x + 1831)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 283*x^12 - 606*x^11 + 1106*x^10 - 1691*x^9 + 2381*x^8 - 3184*x^7 + 3856*x^6 - 3845*x^5 + 5215*x^4 - 6182*x^3 + 1642*x^2 + 1108*x + 1831, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 283 x^{12} - 606 x^{11} + 1106 x^{10} - 1691 x^{9} + 2381 x^{8} - 3184 x^{7} + 3856 x^{6} - 3845 x^{5} + 5215 x^{4} - 6182 x^{3} + 1642 x^{2} + 1108 x + 1831 \)

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:  \(4147643767353759765625=5^{12}\cdot 13^{4}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.44$
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:  $5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{7} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{11} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{18371653321} a^{14} - \frac{1}{2624521903} a^{13} - \frac{1000588186}{18371653321} a^{12} + \frac{754485401}{18371653321} a^{11} - \frac{534250358}{18371653321} a^{10} + \frac{5378382425}{18371653321} a^{9} - \frac{1011493682}{18371653321} a^{8} - \frac{4177848707}{18371653321} a^{7} + \frac{1455735553}{18371653321} a^{6} + \frac{7748190110}{18371653321} a^{5} - \frac{796703273}{2624521903} a^{4} - \frac{7992012373}{18371653321} a^{3} + \frac{9013712543}{18371653321} a^{2} - \frac{1432867906}{18371653321} a + \frac{1533824647}{18371653321}$, $\frac{1}{224115798862879} a^{15} + \frac{6092}{224115798862879} a^{14} - \frac{4709392924861}{224115798862879} a^{13} - \frac{7508576601021}{224115798862879} a^{12} - \frac{12287726235464}{224115798862879} a^{11} + \frac{822317735131}{20374163532989} a^{10} - \frac{66476047108388}{224115798862879} a^{9} + \frac{68993029486695}{224115798862879} a^{8} + \frac{44766085205355}{224115798862879} a^{7} - \frac{11595188029697}{32016542694697} a^{6} + \frac{1526164439078}{20374163532989} a^{5} - \frac{4907823376583}{224115798862879} a^{4} + \frac{26920198625494}{224115798862879} a^{3} - \frac{4554300486304}{32016542694697} a^{2} + \frac{65439335010442}{224115798862879} a + \frac{45872394280395}{224115798862879}$

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

Class group and class number

$C_{6}$, which has order $6$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1790.28536491 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T500):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 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

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.15243125.1, 8.0.64402203125.1, 8.4.2220765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$