↑

Properties

Label 16.0.19112567535...4569.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 43^{12}$
Root discriminant $43.91$
Ramified primes $3, 43$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15625, 24250, 53511, -23638, 20405, 15372, -9998, 8296, 1686, -1352, 1258, 84, -91, 62, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 62*x^13 - 91*x^12 + 84*x^11 + 1258*x^10 - 1352*x^9 + 1686*x^8 + 8296*x^7 - 9998*x^6 + 15372*x^5 + 20405*x^4 - 23638*x^3 + 53511*x^2 + 24250*x + 15625)
 
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 + 62*x^13 - 91*x^12 + 84*x^11 + 1258*x^10 - 1352*x^9 + 1686*x^8 + 8296*x^7 - 9998*x^6 + 15372*x^5 + 20405*x^4 - 23638*x^3 + 53511*x^2 + 24250*x + 15625, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 3 x^{14} + 62 x^{13} - 91 x^{12} + 84 x^{11} + 1258 x^{10} - 1352 x^{9} + 1686 x^{8} + 8296 x^{7} - 9998 x^{6} + 15372 x^{5} + 20405 x^{4} - 23638 x^{3} + 53511 x^{2} + 24250 x + 15625 \)

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:  \(191125675354752187786114569=3^{14}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.91$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{6} - \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a$, $\frac{1}{24} a^{8} - \frac{1}{24} a^{7} + \frac{1}{24} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{5}{24} a^{2} + \frac{5}{24} a - \frac{5}{24}$, $\frac{1}{48} a^{9} - \frac{1}{16} a^{6} - \frac{3}{16} a^{3} - \frac{5}{48}$, $\frac{1}{48} a^{10} - \frac{1}{16} a^{7} - \frac{3}{16} a^{4} - \frac{5}{48} a$, $\frac{1}{192} a^{11} + \frac{1}{192} a^{10} + \frac{1}{192} a^{9} - \frac{1}{64} a^{8} - \frac{1}{64} a^{7} - \frac{1}{64} a^{6} + \frac{5}{64} a^{5} + \frac{5}{64} a^{4} + \frac{5}{64} a^{3} - \frac{77}{192} a^{2} - \frac{77}{192} a - \frac{77}{192}$, $\frac{1}{384} a^{12} - \frac{1}{96} a^{10} - \frac{1}{48} a^{8} - \frac{1}{96} a^{7} - \frac{1}{192} a^{6} + \frac{1}{12} a^{5} - \frac{23}{96} a^{4} - \frac{1}{4} a^{3} - \frac{19}{48} a^{2} + \frac{25}{96} a + \frac{97}{384}$, $\frac{1}{5760} a^{13} - \frac{1}{2880} a^{12} + \frac{1}{960} a^{11} - \frac{7}{2880} a^{10} - \frac{19}{2880} a^{9} - \frac{1}{320} a^{8} + \frac{11}{240} a^{7} - \frac{11}{960} a^{6} + \frac{223}{960} a^{5} - \frac{121}{2880} a^{4} - \frac{397}{2880} a^{3} - \frac{349}{960} a^{2} + \frac{1783}{5760} a + \frac{77}{288}$, $\frac{1}{8939520} a^{14} + \frac{107}{2979840} a^{13} - \frac{1991}{1787904} a^{12} - \frac{4709}{2234880} a^{11} + \frac{511}{148992} a^{10} - \frac{7723}{2234880} a^{9} + \frac{1381}{99328} a^{8} - \frac{88649}{1489920} a^{7} - \frac{12769}{297984} a^{6} + \frac{169213}{2234880} a^{5} + \frac{10841}{148992} a^{4} + \frac{480611}{2234880} a^{3} + \frac{4047901}{8939520} a^{2} - \frac{200377}{2979840} a + \frac{718765}{1787904}$, $\frac{1}{13853350656000} a^{15} - \frac{111559}{4617783552000} a^{14} + \frac{2787901}{4617783552000} a^{13} + \frac{1145916031}{6926675328000} a^{12} + \frac{181047757}{1154445888000} a^{11} + \frac{3492571757}{1154445888000} a^{10} + \frac{52614833429}{6926675328000} a^{9} + \frac{2089031911}{769630592000} a^{8} - \frac{107830571519}{2308891776000} a^{7} - \frac{101584551463}{1731668832000} a^{6} + \frac{2594604443}{11901504000} a^{5} - \frac{14560656469}{1154445888000} a^{4} - \frac{94473090959}{2770670131200} a^{3} + \frac{32735751229}{4617783552000} a^{2} - \frac{1579327292063}{4617783552000} a + \frac{14659877425}{55413402624}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -\frac{134851}{3967168000} a^{15} + \frac{171551}{1983584000} a^{14} - \frac{1147217}{5950752000} a^{13} - \frac{22590961}{11901504000} a^{12} + \frac{5901331}{1487688000} a^{11} - \frac{910787}{123974000} a^{10} - \frac{68468629}{1983584000} a^{9} + \frac{60812163}{991792000} a^{8} - \frac{123867609}{991792000} a^{7} - \frac{333709973}{1983584000} a^{6} + \frac{190780031}{495896000} a^{5} - \frac{266921063}{371922000} a^{4} - \frac{371046053}{2380300800} a^{3} + \frac{4067877457}{5950752000} a^{2} - \frac{3255144243}{1983584000} a + \frac{5991625}{31737344} \) (order $6$)
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:  \( 8934543.27914 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-43}) \), \(\Q(\sqrt{129}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-43})\), 4.2.49923.2 x2, 4.0.1161.1 x2, 8.0.2492305929.1, 8.2.13824820988163.1 x4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
$43$43.8.6.2$x^{8} + 215 x^{4} + 16641$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
43.8.6.2$x^{8} + 215 x^{4} + 16641$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$