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Properties

Label 16.0.11732048713...4128.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 13^{12}\cdot 17^{15}$
Root discriminant $655.90$
Ramified primes $2, 13, 17$
Class number $1249559552$ (GRH)
Class group $[2, 2, 2, 2, 4, 4, 4881092]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2900178534032, 0, 3771231912032, 0, 1370532909096, 0, 113320451504, 0, 3323256560, 0, 44336136, 0, 288626, 0, 884, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 884*x^14 + 288626*x^12 + 44336136*x^10 + 3323256560*x^8 + 113320451504*x^6 + 1370532909096*x^4 + 3771231912032*x^2 + 2900178534032)
 
gp: K = bnfinit(x^16 + 884*x^14 + 288626*x^12 + 44336136*x^10 + 3323256560*x^8 + 113320451504*x^6 + 1370532909096*x^4 + 3771231912032*x^2 + 2900178534032, 1)
 

Normalized defining polynomial

\( x^{16} + 884 x^{14} + 288626 x^{12} + 44336136 x^{10} + 3323256560 x^{8} + 113320451504 x^{6} + 1370532909096 x^{4} + 3771231912032 x^{2} + 2900178534032 \)

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:  \(1173204871331335989075336895013770710512304128=2^{44}\cdot 13^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $655.90$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3536=2^{4}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3536}(1,·)$, $\chi_{3536}(1451,·)$, $\chi_{3536}(1665,·)$, $\chi_{3536}(1481,·)$, $\chi_{3536}(1041,·)$, $\chi_{3536}(2579,·)$, $\chi_{3536}(25,·)$, $\chi_{3536}(915,·)$, $\chi_{3536}(2995,·)$, $\chi_{3536}(827,·)$, $\chi_{3536}(2729,·)$, $\chi_{3536}(619,·)$, $\chi_{3536}(625,·)$, $\chi_{3536}(1331,·)$, $\chi_{3536}(1273,·)$, $\chi_{3536}(1659,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{26} a^{4}$, $\frac{1}{26} a^{5}$, $\frac{1}{26} a^{6}$, $\frac{1}{26} a^{7}$, $\frac{1}{676} a^{8}$, $\frac{1}{676} a^{9}$, $\frac{1}{8788} a^{10} - \frac{3}{169} a^{6} - \frac{3}{13} a^{2}$, $\frac{1}{8788} a^{11} - \frac{3}{169} a^{7} - \frac{3}{13} a^{3}$, $\frac{1}{70304} a^{12} - \frac{1}{52} a^{6} + \frac{1}{4}$, $\frac{1}{3304288} a^{13} - \frac{1}{103259} a^{11} + \frac{3}{31772} a^{9} + \frac{373}{31772} a^{7} + \frac{1}{611} a^{5} - \frac{66}{611} a^{3} - \frac{59}{188} a$, $\frac{1}{23124844883412370472052827608928} a^{14} + \frac{3683314321900718365186591}{1778834221800951574773294431456} a^{12} + \frac{6646043831936545145246177}{222354277725118946846661803932} a^{10} - \frac{3152987177061367171599231}{8552087604812267186410069382} a^{8} + \frac{87187119058305417583838901}{17104175209624534372820138764} a^{6} + \frac{6034214869482354989911947}{328926446338933353323464207} a^{4} + \frac{317097657310376683391524961}{1315705785355733413293856828} a^{2} + \frac{997508600856513647014921}{2153364624150136519302548}$, $\frac{1}{300622983484360816136686758916064} a^{15} + \frac{2606632009825650105535317}{23124844883412370472052827608928} a^{13} - \frac{90255364254819598223368483}{2890605610426546309006603451116} a^{11} - \frac{80674113646466281866490959}{111177138862559473423330901966} a^{9} + \frac{4189346728064315486855192841}{222354277725118946846661803932} a^{7} + \frac{381348658262098979250821}{328926446338933353323464207} a^{5} + \frac{500133650363138287532241541}{17104175209624534372820138764} a^{3} - \frac{30433348155584377946690179}{101208137335056416407219756} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4881092}$, which has order $1249559552$ (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:  \( 12339810.435594495 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.48003820910809088.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.22.30$x^{8} + 2 x^{4} + 16 x^{3} + 16 x + 52$$4$$2$$22$$C_8$$[3, 4]^{2}$
2.8.22.30$x^{8} + 2 x^{4} + 16 x^{3} + 16 x + 52$$4$$2$$22$$C_8$$[3, 4]^{2}$
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
17Data not computed