↑

Properties

Label 16.12.6690022700...9293.1
Degree $16$
Signature $[12, 2]$
Discriminant $3^{8}\cdot 13^{9}\cdot 1327^{8}$
Root discriminant $267.05$
Ramified primes $3, 13, 1327$
Class number $48$ (GRH)
Class group $[48]$ (GRH)
Galois group 16T1675

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1096054873, -1333911191, -10433250, 701260374, -196639436, -136010120, 45769170, 11770693, -3864589, -394416, 119099, -361, 609, 210, -87, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 87*x^14 + 210*x^13 + 609*x^12 - 361*x^11 + 119099*x^10 - 394416*x^9 - 3864589*x^8 + 11770693*x^7 + 45769170*x^6 - 136010120*x^5 - 196639436*x^4 + 701260374*x^3 - 10433250*x^2 - 1333911191*x + 1096054873)
 
gp: K = bnfinit(x^16 - 2*x^15 - 87*x^14 + 210*x^13 + 609*x^12 - 361*x^11 + 119099*x^10 - 394416*x^9 - 3864589*x^8 + 11770693*x^7 + 45769170*x^6 - 136010120*x^5 - 196639436*x^4 + 701260374*x^3 - 10433250*x^2 - 1333911191*x + 1096054873, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 87 x^{14} + 210 x^{13} + 609 x^{12} - 361 x^{11} + 119099 x^{10} - 394416 x^{9} - 3864589 x^{8} + 11770693 x^{7} + 45769170 x^{6} - 136010120 x^{5} - 196639436 x^{4} + 701260374 x^{3} - 10433250 x^{2} - 1333911191 x + 1096054873 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(669002270083080728394108412888785769293=3^{8}\cdot 13^{9}\cdot 1327^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $267.05$
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:  $3, 13, 1327$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{561150406222600917531075447042183930391970228633364878020081} a^{15} - \frac{194903308902141772522521784613184191269134561484293103834298}{561150406222600917531075447042183930391970228633364878020081} a^{14} - \frac{240236535417467931004580845481557319400783122856706176493395}{561150406222600917531075447042183930391970228633364878020081} a^{13} + \frac{154489026398451282836587576377594005024769951262621852585114}{561150406222600917531075447042183930391970228633364878020081} a^{12} + \frac{141942586753046214190507464389286611918320543873165340631773}{561150406222600917531075447042183930391970228633364878020081} a^{11} + \frac{47152436808465046198633920869008692452119802367316990682478}{561150406222600917531075447042183930391970228633364878020081} a^{10} - \frac{3635227381481930265730619624269256178252702228670995298684}{8375379197352252500463812642420655677492092964677087731643} a^{9} + \frac{9570238211663002993056553233652357413079972533289550229702}{561150406222600917531075447042183930391970228633364878020081} a^{8} + \frac{220851905769825143437856091672892534094658190596916188927989}{561150406222600917531075447042183930391970228633364878020081} a^{7} - \frac{3311907859255324127481838940229283004582641868615202276101}{10587743513633979576058027302682715667773023181761601472077} a^{6} - \frac{200523671365519853223763785353913007050927220330385934938656}{561150406222600917531075447042183930391970228633364878020081} a^{5} + \frac{268103409606152701538263479103338297524299134951209183811819}{561150406222600917531075447042183930391970228633364878020081} a^{4} - \frac{93384983659645845611617226820433871630084209114483265882794}{561150406222600917531075447042183930391970228633364878020081} a^{3} + \frac{72521972890317022668271811967805459694310869344280719659176}{561150406222600917531075447042183930391970228633364878020081} a^{2} - \frac{264077145291660799675564089130360674681742001636389226723531}{561150406222600917531075447042183930391970228633364878020081} a + \frac{35716507986041589932527696748371116167238102368184145830910}{561150406222600917531075447042183930391970228633364878020081}$

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

Class group and class number

$C_{48}$, 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:  $13$
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:  \( 1486109476920 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1675:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 54 conjugacy class representatives for t16n1675 are not computed
Character table for t16n1675 is not computed

Intermediate fields

\(\Q(\sqrt{51753}) \), 4.4.3981.1, 8.8.7173681975339714081.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1327Data not computed