↑

Properties

Label 16.0.98586762102...169.10
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 157^{6}$
Root discriminant $99.91$
Ramified primes $37, 157$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1263

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![39371633, -52490006, 50035370, -19749908, 2080962, 4196569, -2139347, 248849, 274990, -125210, 24860, -2766, 880, -108, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + x^14 - 108*x^13 + 880*x^12 - 2766*x^11 + 24860*x^10 - 125210*x^9 + 274990*x^8 + 248849*x^7 - 2139347*x^6 + 4196569*x^5 + 2080962*x^4 - 19749908*x^3 + 50035370*x^2 - 52490006*x + 39371633)
 
gp: K = bnfinit(x^16 - 3*x^15 + x^14 - 108*x^13 + 880*x^12 - 2766*x^11 + 24860*x^10 - 125210*x^9 + 274990*x^8 + 248849*x^7 - 2139347*x^6 + 4196569*x^5 + 2080962*x^4 - 19749908*x^3 + 50035370*x^2 - 52490006*x + 39371633, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + x^{14} - 108 x^{13} + 880 x^{12} - 2766 x^{11} + 24860 x^{10} - 125210 x^{9} + 274990 x^{8} + 248849 x^{7} - 2139347 x^{6} + 4196569 x^{5} + 2080962 x^{4} - 19749908 x^{3} + 50035370 x^{2} - 52490006 x + 39371633 \)

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:  \(98586762102441645314452445352169=37^{12}\cdot 157^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.91$
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:  $37, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{21} a^{14} - \frac{1}{21} a^{13} - \frac{4}{21} a^{11} - \frac{10}{21} a^{10} + \frac{1}{7} a^{9} - \frac{8}{21} a^{8} + \frac{8}{21} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{8}{21} a^{3} - \frac{4}{21} a^{2} + \frac{3}{7} a + \frac{1}{3}$, $\frac{1}{189226907151976520223224232345710226923083947709400540583} a^{15} + \frac{4325124873603131798913105136762630385096878298064879986}{189226907151976520223224232345710226923083947709400540583} a^{14} + \frac{3198583282037515954227996965802132408787445551194607070}{63075635717325506741074744115236742307694649236466846861} a^{13} - \frac{2306994858933501602781468467307490930333955820703314352}{189226907151976520223224232345710226923083947709400540583} a^{12} + \frac{65519669250446245115362351244701064294300042003243447162}{189226907151976520223224232345710226923083947709400540583} a^{11} + \frac{14473602426641763352238940349176514067470281475644760694}{63075635717325506741074744115236742307694649236466846861} a^{10} + \frac{40371281519364740770793466772367573561777222895092369110}{189226907151976520223224232345710226923083947709400540583} a^{9} - \frac{17192837697688456044353234574232056205442913775267786953}{63075635717325506741074744115236742307694649236466846861} a^{8} + \frac{31948565258121158065600572251099826199275126539657540120}{189226907151976520223224232345710226923083947709400540583} a^{7} - \frac{5447281143901899801982515009805069490537612308573266405}{63075635717325506741074744115236742307694649236466846861} a^{6} - \frac{2603206550152594926849725231925946256826845047012849791}{63075635717325506741074744115236742307694649236466846861} a^{5} - \frac{22284347111592542401667351478763808278730374078608795038}{189226907151976520223224232345710226923083947709400540583} a^{4} - \frac{53858421998399380274496062905385178536068305266218280935}{189226907151976520223224232345710226923083947709400540583} a^{3} + \frac{375503758138061766959820361464661206879346740303722333}{1342034802496287377469675406707164729950949983754613763} a^{2} - \frac{3264100019496528663917365174079424910142233820364858236}{189226907151976520223224232345710226923083947709400540583} a - \frac{6018118964135862067642732862408702369307606209730364}{170015190612737214935511439663710895708071830826056191}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2121577169.95 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 8.0.402819046213.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
157Data not computed