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Properties

Label 16.0.36086878424...2857.2
Degree $16$
Signature $[0, 8]$
Discriminant $29^{15}\cdot 53^{5}$
Root discriminant $81.25$
Ramified primes $29, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4856944, 1342760, 824640, -5494616, 71005, 1961670, 237830, -642987, 70664, 65836, -11651, -3500, 840, 143, -38, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 38*x^14 + 143*x^13 + 840*x^12 - 3500*x^11 - 11651*x^10 + 65836*x^9 + 70664*x^8 - 642987*x^7 + 237830*x^6 + 1961670*x^5 + 71005*x^4 - 5494616*x^3 + 824640*x^2 + 1342760*x + 4856944)
 
gp: K = bnfinit(x^16 - 2*x^15 - 38*x^14 + 143*x^13 + 840*x^12 - 3500*x^11 - 11651*x^10 + 65836*x^9 + 70664*x^8 - 642987*x^7 + 237830*x^6 + 1961670*x^5 + 71005*x^4 - 5494616*x^3 + 824640*x^2 + 1342760*x + 4856944, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 38 x^{14} + 143 x^{13} + 840 x^{12} - 3500 x^{11} - 11651 x^{10} + 65836 x^{9} + 70664 x^{8} - 642987 x^{7} + 237830 x^{6} + 1961670 x^{5} + 71005 x^{4} - 5494616 x^{3} + 824640 x^{2} + 1342760 x + 4856944 \)

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:  \(3608687842491875308187596442857=29^{15}\cdot 53^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.25$
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:  $29, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{39} a^{12} - \frac{2}{39} a^{11} + \frac{6}{13} a^{10} - \frac{8}{39} a^{9} - \frac{2}{39} a^{8} - \frac{14}{39} a^{7} - \frac{6}{13} a^{6} + \frac{19}{39} a^{5} - \frac{1}{13} a^{4} - \frac{6}{13} a^{3} - \frac{10}{39} a^{2} + \frac{4}{39} a - \frac{16}{39}$, $\frac{1}{390} a^{13} + \frac{1}{195} a^{12} + \frac{44}{195} a^{11} + \frac{103}{390} a^{10} - \frac{19}{39} a^{9} - \frac{10}{39} a^{8} - \frac{7}{78} a^{7} - \frac{7}{195} a^{6} - \frac{61}{195} a^{5} - \frac{49}{130} a^{4} - \frac{41}{195} a^{3} + \frac{7}{65} a^{2} - \frac{3}{10} a - \frac{71}{195}$, $\frac{1}{50700} a^{14} + \frac{2}{4225} a^{13} + \frac{7}{8450} a^{12} - \frac{11821}{50700} a^{11} + \frac{1441}{8450} a^{10} - \frac{334}{2535} a^{9} - \frac{1437}{3380} a^{8} - \frac{10487}{25350} a^{7} - \frac{1052}{2535} a^{6} + \frac{12229}{50700} a^{5} - \frac{3589}{12675} a^{4} - \frac{5531}{25350} a^{3} - \frac{3981}{16900} a^{2} + \frac{4897}{25350} a - \frac{2956}{12675}$, $\frac{1}{134273886061627311831947467053827162583000} a^{15} - \frac{13803548144149408976391165579260153}{16784235757703413978993433381728395322875} a^{14} + \frac{2026522675676009800340320794679047737}{1918198372308961597599249529340388036900} a^{13} + \frac{433285904215196040539452148406873347921}{44757962020542437277315822351275720861000} a^{12} + \frac{7028479658904216745228750520459800526709}{22378981010271218638657911175637860430500} a^{11} - \frac{3721356709624302835435567161930276147149}{11189490505135609319328955587818930215250} a^{10} + \frac{2814399036304159690644670282518317940579}{8951592404108487455463164470255144172200} a^{9} + \frac{411481858180560312409591723848363486923}{3196997287181602662665415882233980061500} a^{8} + \frac{628284478915705969052910322012174955203}{1291095058284877998384110260132953486375} a^{7} - \frac{60439658524763704536132465316064111116451}{134273886061627311831947467053827162583000} a^{6} + \frac{1083244356427772433714898305735197426123}{5594745252567804659664477793909465107625} a^{5} + \frac{9921095143561791636384793304586690730571}{22378981010271218638657911175637860430500} a^{4} + \frac{55900095995595624326965401118237513949033}{134273886061627311831947467053827162583000} a^{3} - \frac{3696084252602608271764393922091787702553}{9590991861544807987996247646701940184500} a^{2} + \frac{185591963424321650345384436942183869422}{430365019428292666128036753377651162125} a - \frac{7320599551486095059194604074710711629756}{16784235757703413978993433381728395322875}$

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:  \( 390093291.819 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 8.0.914243444377.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
29Data not computed
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$