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Properties

Label 16.0.25916019301...625.11
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{6}$
Root discriminant $18.87$
Ramified primes $5, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4111, -9802, 16675, -16596, 12348, -4451, -196, 2156, -1185, 202, 478, -456, 292, -116, 37, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 37*x^14 - 116*x^13 + 292*x^12 - 456*x^11 + 478*x^10 + 202*x^9 - 1185*x^8 + 2156*x^7 - 196*x^6 - 4451*x^5 + 12348*x^4 - 16596*x^3 + 16675*x^2 - 9802*x + 4111)
 
gp: K = bnfinit(x^16 - 7*x^15 + 37*x^14 - 116*x^13 + 292*x^12 - 456*x^11 + 478*x^10 + 202*x^9 - 1185*x^8 + 2156*x^7 - 196*x^6 - 4451*x^5 + 12348*x^4 - 16596*x^3 + 16675*x^2 - 9802*x + 4111, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 37 x^{14} - 116 x^{13} + 292 x^{12} - 456 x^{11} + 478 x^{10} + 202 x^{9} - 1185 x^{8} + 2156 x^{7} - 196 x^{6} - 4451 x^{5} + 12348 x^{4} - 16596 x^{3} + 16675 x^{2} - 9802 x + 4111 \)

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:  \(259160193017822265625=5^{12}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.87$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{2} a^{9} + \frac{2}{5} a^{8} - \frac{3}{10} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{10} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{10} + \frac{2}{5} a^{9} + \frac{3}{10} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} - \frac{3}{10} a^{3} - \frac{1}{10} a^{2} + \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{110} a^{14} - \frac{1}{22} a^{13} + \frac{3}{110} a^{12} - \frac{23}{110} a^{11} - \frac{13}{110} a^{10} + \frac{24}{55} a^{9} + \frac{17}{110} a^{8} - \frac{3}{110} a^{7} - \frac{13}{55} a^{6} - \frac{13}{110} a^{5} + \frac{5}{11} a^{4} + \frac{31}{110} a^{3} - \frac{21}{110} a^{2} - \frac{23}{55} a - \frac{7}{110}$, $\frac{1}{52050720885187090} a^{15} + \frac{4400931256093}{1679055512425390} a^{14} + \frac{792081687811294}{26025360442593545} a^{13} + \frac{4792396633925}{253905955537498} a^{12} + \frac{419406704197037}{4731883716835190} a^{11} - \frac{5677047039232297}{52050720885187090} a^{10} - \frac{4184760546996229}{52050720885187090} a^{9} - \frac{1271254152147608}{26025360442593545} a^{8} - \frac{1806470072237786}{26025360442593545} a^{7} + \frac{6496138528363017}{52050720885187090} a^{6} + \frac{3681315408192834}{26025360442593545} a^{5} + \frac{1950210787064059}{5205072088518709} a^{4} - \frac{7636224489695101}{26025360442593545} a^{3} + \frac{1913258341488887}{10410144177037418} a^{2} + \frac{177918561870228}{5205072088518709} a + \frac{11279759223553234}{26025360442593545}$

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:  \( -\frac{26902112308265}{5205072088518709} a^{15} + \frac{33926820817163}{839527756212695} a^{14} - \frac{5464094184343431}{26025360442593545} a^{13} + \frac{7934920372712}{11541179797159} a^{12} - \frac{43048132468419363}{26025360442593545} a^{11} + \frac{68013552511812938}{26025360442593545} a^{10} - \frac{1001993587265814}{473188371683519} a^{9} - \frac{49492861378153158}{26025360442593545} a^{8} + \frac{232195811428391898}{26025360442593545} a^{7} - \frac{295763203487827842}{26025360442593545} a^{6} - \frac{19981593167055989}{26025360442593545} a^{5} + \frac{82632287872460903}{2365941858417595} a^{4} - \frac{370497867576183199}{5205072088518709} a^{3} + \frac{2313446552386777309}{26025360442593545} a^{2} - \frac{1661719226709335379}{26025360442593545} a + \frac{773219108489750616}{26025360442593545} \) (order $10$)
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:  \( 8414.57802499 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.1578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed