↑

Properties

Label 16.8.85977968400...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 181^{2}$
Root discriminant $27.13$
Ramified primes $2, 3, 5, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1102

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 48, -200, -80, 960, -836, -628, 1660, -1395, 360, 412, -524, 330, -140, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 330*x^12 - 524*x^11 + 412*x^10 + 360*x^9 - 1395*x^8 + 1660*x^7 - 628*x^6 - 836*x^5 + 960*x^4 - 80*x^3 - 200*x^2 + 48*x + 16)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 330*x^12 - 524*x^11 + 412*x^10 + 360*x^9 - 1395*x^8 + 1660*x^7 - 628*x^6 - 836*x^5 + 960*x^4 - 80*x^3 - 200*x^2 + 48*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 140 x^{13} + 330 x^{12} - 524 x^{11} + 412 x^{10} + 360 x^{9} - 1395 x^{8} + 1660 x^{7} - 628 x^{6} - 836 x^{5} + 960 x^{4} - 80 x^{3} - 200 x^{2} + 48 x + 16 \)

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:  \(85977968400000000000000=2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.13$
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:  $2, 3, 5, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} - \frac{1}{4} a^{4} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{20} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{3}{20} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} + \frac{1}{10} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{10} - \frac{1}{4} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{20} a^{11} - \frac{1}{4} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} - \frac{1}{40} a^{8} + \frac{1}{5} a^{7} - \frac{7}{40} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{40} a^{13} - \frac{1}{40} a^{11} - \frac{1}{40} a^{9} + \frac{1}{8} a^{7} - \frac{1}{5} a^{6} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{36200} a^{14} - \frac{7}{36200} a^{13} + \frac{259}{36200} a^{12} + \frac{347}{36200} a^{11} - \frac{519}{36200} a^{10} - \frac{451}{36200} a^{9} + \frac{317}{36200} a^{8} - \frac{1699}{36200} a^{7} + \frac{4409}{18100} a^{6} + \frac{4207}{18100} a^{5} + \frac{1961}{9050} a^{4} - \frac{697}{9050} a^{3} + \frac{1902}{4525} a^{2} + \frac{56}{4525} a + \frac{913}{4525}$, $\frac{1}{36200} a^{15} + \frac{21}{3620} a^{13} + \frac{7}{724} a^{12} + \frac{1}{362} a^{11} - \frac{58}{4525} a^{10} + \frac{39}{1810} a^{9} + \frac{13}{905} a^{8} + \frac{471}{7240} a^{7} - \frac{147}{905} a^{6} - \frac{1019}{18100} a^{5} - \frac{399}{3620} a^{4} + \frac{69}{181} a^{3} - \frac{41}{905} a^{2} + \frac{261}{905} a + \frac{961}{4525}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 282983.732049 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 64 conjugacy class representatives for t16n1102 are not computed
Character table for t16n1102 is not computed

Intermediate fields

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
181Data not computed