↑

Properties

Label 16.0.97316019886...9849.3
Degree $16$
Signature $[0, 8]$
Discriminant $37^{6}\cdot 41^{14}$
Root discriminant $99.83$
Ramified primes $37, 41$
Class number $48$ (GRH)
Class group $[2, 24]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4983232, 3217664, 22316704, -13141776, -11557040, 6216884, 2004779, -1258788, -109229, 139988, -5925, -8804, 1089, 292, -55, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 55*x^14 + 292*x^13 + 1089*x^12 - 8804*x^11 - 5925*x^10 + 139988*x^9 - 109229*x^8 - 1258788*x^7 + 2004779*x^6 + 6216884*x^5 - 11557040*x^4 - 13141776*x^3 + 22316704*x^2 + 3217664*x + 4983232)
 
gp: K = bnfinit(x^16 - 4*x^15 - 55*x^14 + 292*x^13 + 1089*x^12 - 8804*x^11 - 5925*x^10 + 139988*x^9 - 109229*x^8 - 1258788*x^7 + 2004779*x^6 + 6216884*x^5 - 11557040*x^4 - 13141776*x^3 + 22316704*x^2 + 3217664*x + 4983232, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 55 x^{14} + 292 x^{13} + 1089 x^{12} - 8804 x^{11} - 5925 x^{10} + 139988 x^{9} - 109229 x^{8} - 1258788 x^{7} + 2004779 x^{6} + 6216884 x^{5} - 11557040 x^{4} - 13141776 x^{3} + 22316704 x^{2} + 3217664 x + 4983232 \)

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:  \(97316019886955839743696883969849=37^{6}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.83$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{1}{4} a^{3} + \frac{3}{10} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{20} a^{4} + \frac{3}{10} a^{3} + \frac{9}{20} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{10} - \frac{1}{40} a^{9} + \frac{1}{20} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{3}{40} a^{5} - \frac{3}{40} a^{4} - \frac{7}{40} a^{3} - \frac{9}{20} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{9} + \frac{1}{20} a^{8} - \frac{1}{10} a^{7} + \frac{7}{40} a^{6} - \frac{1}{5} a^{4} + \frac{17}{40} a^{3} + \frac{3}{20} a^{2} + \frac{1}{10} a + \frac{1}{5}$, $\frac{1}{400} a^{13} + \frac{1}{100} a^{12} + \frac{3}{400} a^{11} - \frac{3}{200} a^{10} - \frac{1}{80} a^{9} + \frac{3}{100} a^{8} - \frac{53}{400} a^{7} + \frac{19}{100} a^{6} - \frac{31}{400} a^{5} - \frac{1}{200} a^{4} + \frac{161}{400} a^{3} - \frac{31}{100} a^{2} + \frac{11}{25} a - \frac{1}{50}$, $\frac{1}{800} a^{14} - \frac{3}{800} a^{12} - \frac{1}{100} a^{11} + \frac{9}{800} a^{10} + \frac{3}{200} a^{9} - \frac{61}{800} a^{8} + \frac{1}{100} a^{7} + \frac{7}{32} a^{6} + \frac{19}{100} a^{5} - \frac{141}{800} a^{4} + \frac{33}{200} a^{3} - \frac{31}{100} a^{2} - \frac{49}{100} a - \frac{4}{25}$, $\frac{1}{3014844846865858439028847026060180800} a^{15} - \frac{13060390386304794775827218606037}{47106950732279038109825734782190325} a^{14} + \frac{1672444299271610713152296416783221}{3014844846865858439028847026060180800} a^{13} - \frac{1503583430052059690635429267414771}{376855605858232304878605878257522600} a^{12} - \frac{25577002113441862447215821086291}{120593793874634337561153881042407232} a^{11} + \frac{39300800346225408199656163608164}{47106950732279038109825734782190325} a^{10} - \frac{56568362244911034338879526349127257}{3014844846865858439028847026060180800} a^{9} + \frac{31597440056048274227825001869425643}{376855605858232304878605878257522600} a^{8} - \frac{110018284850164897315730156253350121}{3014844846865858439028847026060180800} a^{7} + \frac{27331156914760758436119247244880237}{376855605858232304878605878257522600} a^{6} + \frac{340644135756820476130460687892591799}{3014844846865858439028847026060180800} a^{5} - \frac{3389709043252930646058490787649057}{47106950732279038109825734782190325} a^{4} - \frac{696271267561212783374611775471449}{1463516915954300213120799527213680} a^{3} - \frac{93493421068182463082017031561554353}{376855605858232304878605878257522600} a^{2} - \frac{17075716346674758521383441321904912}{47106950732279038109825734782190325} a - \frac{25986572927916708957296922847465349}{94213901464558076219651469564380650}$

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

Class group and class number

$C_{2}\times C_{24}$, which has order $48$ (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:  \( 514989963.509 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.0.2550077.1, 4.0.62197.1, 8.4.266618600943089.1 x2, 8.0.6502892705929.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$