↑

Properties

Label 16.0.86589581483...173.16
Degree $16$
Signature $[0, 8]$
Discriminant $13^{7}\cdot 53^{14}$
Root discriminant $99.10$
Ramified primes $13, 53$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1281

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51239197, -60725949, 70841602, -9106744, -7251219, -1546998, 1574742, 143486, -26277, -39994, 5581, 979, 483, -102, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 102*x^13 + 483*x^12 + 979*x^11 + 5581*x^10 - 39994*x^9 - 26277*x^8 + 143486*x^7 + 1574742*x^6 - 1546998*x^5 - 7251219*x^4 - 9106744*x^3 + 70841602*x^2 - 60725949*x + 51239197)
 
gp: K = bnfinit(x^16 - x^15 - 102*x^13 + 483*x^12 + 979*x^11 + 5581*x^10 - 39994*x^9 - 26277*x^8 + 143486*x^7 + 1574742*x^6 - 1546998*x^5 - 7251219*x^4 - 9106744*x^3 + 70841602*x^2 - 60725949*x + 51239197, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 102 x^{13} + 483 x^{12} + 979 x^{11} + 5581 x^{10} - 39994 x^{9} - 26277 x^{8} + 143486 x^{7} + 1574742 x^{6} - 1546998 x^{5} - 7251219 x^{4} - 9106744 x^{3} + 70841602 x^{2} - 60725949 x + 51239197 \)

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:  \(86589581483779140722048687468173=13^{7}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.10$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} + \frac{1}{13} a^{12} + \frac{5}{13} a^{11} - \frac{3}{13} a^{10} + \frac{6}{13} a^{9} - \frac{5}{13} a^{8} + \frac{1}{13} a^{7} + \frac{3}{13} a^{6} + \frac{4}{13} a^{5} + \frac{4}{13} a^{4} + \frac{5}{13} a^{3} + \frac{4}{13} a^{2} + \frac{2}{13} a - \frac{6}{13}$, $\frac{1}{13} a^{14} + \frac{4}{13} a^{12} + \frac{5}{13} a^{11} - \frac{4}{13} a^{10} + \frac{2}{13} a^{9} + \frac{6}{13} a^{8} + \frac{2}{13} a^{7} + \frac{1}{13} a^{6} + \frac{1}{13} a^{4} - \frac{1}{13} a^{3} - \frac{2}{13} a^{2} + \frac{5}{13} a + \frac{6}{13}$, $\frac{1}{170622510485222938405441027861352780924706924578058791231} a^{15} - \frac{4319366398149451696079405411764271669369617938645350164}{170622510485222938405441027861352780924706924578058791231} a^{14} + \frac{4267901033040968419751085280098977799315172640813498937}{170622510485222938405441027861352780924706924578058791231} a^{13} - \frac{11895493567437071819128169792664282381716168588438376642}{170622510485222938405441027861352780924706924578058791231} a^{12} - \frac{20723202220681305040106893302409734102434179892270639475}{170622510485222938405441027861352780924706924578058791231} a^{11} - \frac{79330458205052069781917294193170437834262531827923804075}{170622510485222938405441027861352780924706924578058791231} a^{10} - \frac{44692748283550120183740646457585186934114893899143324695}{170622510485222938405441027861352780924706924578058791231} a^{9} + \frac{1525300822312093664317018065917385547122825632098649568}{3630266180536658263945553784284101721802274991022527473} a^{8} + \frac{62839845574736518395983340324504333816651006227826319185}{170622510485222938405441027861352780924706924578058791231} a^{7} - \frac{136650403922591146736120049577944408896272081233157459}{13124808498863302954264694450873290840362071121389137787} a^{6} + \frac{4940161953359553146245420806531712950955592645195064349}{13124808498863302954264694450873290840362071121389137787} a^{5} - \frac{75596421013521414887378370303370197177766133963712489854}{170622510485222938405441027861352780924706924578058791231} a^{4} + \frac{56669632335512630612648694514261004639396482233174892125}{170622510485222938405441027861352780924706924578058791231} a^{3} + \frac{35565953044500164558584269336385276942933635532201058453}{170622510485222938405441027861352780924706924578058791231} a^{2} + \frac{4937658655410544496793388754463277763478502855530014982}{170622510485222938405441027861352780924706924578058791231} a + \frac{31342566504905854738422137543650151374290192307674622463}{170622510485222938405441027861352780924706924578058791231}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 272231233.115 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

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 t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.3745777030801.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$