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Properties

Label 16.0.26252865470...9681.6
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 101^{12}$
Root discriminant $218.12$
Ramified primes $13, 101$
Class number $6760$ (GRH)
Class group $[26, 260]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6194357651, -6992958031, 4103046553, -1506235284, 510117913, -174630340, 51922689, -11794245, 2090613, -518319, 104686, -13622, 2319, -334, 57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 57*x^14 - 334*x^13 + 2319*x^12 - 13622*x^11 + 104686*x^10 - 518319*x^9 + 2090613*x^8 - 11794245*x^7 + 51922689*x^6 - 174630340*x^5 + 510117913*x^4 - 1506235284*x^3 + 4103046553*x^2 - 6992958031*x + 6194357651)
 
gp: K = bnfinit(x^16 + 57*x^14 - 334*x^13 + 2319*x^12 - 13622*x^11 + 104686*x^10 - 518319*x^9 + 2090613*x^8 - 11794245*x^7 + 51922689*x^6 - 174630340*x^5 + 510117913*x^4 - 1506235284*x^3 + 4103046553*x^2 - 6992958031*x + 6194357651, 1)
 

Normalized defining polynomial

\( x^{16} + 57 x^{14} - 334 x^{13} + 2319 x^{12} - 13622 x^{11} + 104686 x^{10} - 518319 x^{9} + 2090613 x^{8} - 11794245 x^{7} + 51922689 x^{6} - 174630340 x^{5} + 510117913 x^{4} - 1506235284 x^{3} + 4103046553 x^{2} - 6992958031 x + 6194357651 \)

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:  \(26252865470156848284984700456389559681=13^{12}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $218.12$
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:  $13, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{1716} a^{12} + \frac{115}{572} a^{11} + \frac{43}{858} a^{10} + \frac{5}{286} a^{9} - \frac{49}{286} a^{8} - \frac{135}{572} a^{7} - \frac{673}{1716} a^{6} - \frac{7}{22} a^{5} + \frac{799}{1716} a^{4} + \frac{353}{1716} a^{3} + \frac{137}{572} a^{2} - \frac{19}{156} a - \frac{337}{1716}$, $\frac{1}{1716} a^{13} + \frac{323}{1716} a^{11} + \frac{5}{22} a^{10} - \frac{29}{143} a^{9} - \frac{73}{572} a^{8} - \frac{401}{858} a^{7} + \frac{279}{572} a^{6} - \frac{449}{1716} a^{5} - \frac{371}{858} a^{4} + \frac{7}{26} a^{3} - \frac{217}{858} a^{2} - \frac{76}{429} a - \frac{141}{572}$, $\frac{1}{1716} a^{14} - \frac{11}{52} a^{11} + \frac{47}{429} a^{10} + \frac{129}{572} a^{9} - \frac{5}{39} a^{8} + \frac{63}{286} a^{7} - \frac{12}{143} a^{6} - \frac{137}{858} a^{5} - \frac{215}{1716} a^{4} + \frac{173}{572} a^{3} - \frac{67}{1716} a^{2} + \frac{40}{429} a + \frac{743}{1716}$, $\frac{1}{1024419894768838833067546270052623069054908815383195931844} a^{15} - \frac{75763534682395839006876504070845363421346377366062689}{341473298256279611022515423350874356351636271794398643948} a^{14} - \frac{297938926154959453432347070085739881606298226934556431}{1024419894768838833067546270052623069054908815383195931844} a^{13} + \frac{899373480928672500771675074118757985788083670651607}{93129081342621712097049660913874824459537165034835993804} a^{12} - \frac{106995321117931185426893963059175244763261183077259633555}{512209947384419416533773135026311534527454407691597965922} a^{11} + \frac{37422150038775234723224931332030802493996016992689142601}{1024419894768838833067546270052623069054908815383195931844} a^{10} - \frac{114528270160405521420135170480897867119227266412695212447}{1024419894768838833067546270052623069054908815383195931844} a^{9} - \frac{30454705399052618096049124411089684171226327531650773285}{341473298256279611022515423350874356351636271794398643948} a^{8} - \frac{14048386547344054273243475734146620104115194215347623853}{39400765183416878194905625771254733425188800591661381994} a^{7} - \frac{498838885513976010935434902830056455449245709562152064737}{1024419894768838833067546270052623069054908815383195931844} a^{6} - \frac{20472637316749316187003365364813466384974022137186177857}{512209947384419416533773135026311534527454407691597965922} a^{5} + \frac{8132528540893257711766321283296011084256990603146858937}{170736649128139805511257711675437178175818135897199321974} a^{4} - \frac{244292409438286526695965035745271755738220954912132206631}{512209947384419416533773135026311534527454407691597965922} a^{3} - \frac{162246474923720584873657014933018875810592116031816552985}{1024419894768838833067546270052623069054908815383195931844} a^{2} - \frac{287297167500256771563309579547781994741410287418874026161}{1024419894768838833067546270052623069054908815383195931844} a + \frac{1838598700782046167669999767360103551217875377292030087}{4787008854059994547044608738563659201191162688706523046}$

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

Class group and class number

$C_{26}\times C_{260}$, which has order $6760$ (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:  \( 151720134.33 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{1313}) \), \(\Q(\sqrt{13}, \sqrt{101})\), 4.0.13393913.1 x2, 4.0.174120869.1 x2, 4.0.22411597.2, 4.0.2197.1, 8.0.30318077021315161.3, 8.0.502279680090409.5, 8.8.5123755016602262209.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$