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Properties

Label 16.16.2625286547...9681.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{12}\cdot 101^{12}$
Root discriminant $218.12$
Ramified primes $13, 101$
Class number $1$ (GRH)
Class group Trivial (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![-53337296, -345202792, -276793968, 466800116, 384304713, -223518780, -135765373, 38672703, 18517725, -2853491, -1157913, 90172, 33397, -978, -386, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 386*x^14 - 978*x^13 + 33397*x^12 + 90172*x^11 - 1157913*x^10 - 2853491*x^9 + 18517725*x^8 + 38672703*x^7 - 135765373*x^6 - 223518780*x^5 + 384304713*x^4 + 466800116*x^3 - 276793968*x^2 - 345202792*x - 53337296)
 
gp: K = bnfinit(x^16 - x^15 - 386*x^14 - 978*x^13 + 33397*x^12 + 90172*x^11 - 1157913*x^10 - 2853491*x^9 + 18517725*x^8 + 38672703*x^7 - 135765373*x^6 - 223518780*x^5 + 384304713*x^4 + 466800116*x^3 - 276793968*x^2 - 345202792*x - 53337296, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 386 x^{14} - 978 x^{13} + 33397 x^{12} + 90172 x^{11} - 1157913 x^{10} - 2853491 x^{9} + 18517725 x^{8} + 38672703 x^{7} - 135765373 x^{6} - 223518780 x^{5} + 384304713 x^{4} + 466800116 x^{3} - 276793968 x^{2} - 345202792 x - 53337296 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{34} a^{10} - \frac{2}{17} a^{9} + \frac{13}{34} a^{8} - \frac{7}{34} a^{7} - \frac{4}{17} a^{6} - \frac{6}{17} a^{5} + \frac{3}{34} a^{4} + \frac{13}{34} a^{3} + \frac{1}{17} a^{2} - \frac{13}{34} a$, $\frac{1}{34} a^{11} - \frac{3}{34} a^{9} + \frac{11}{34} a^{8} - \frac{1}{17} a^{7} - \frac{5}{17} a^{6} - \frac{11}{34} a^{5} - \frac{9}{34} a^{4} - \frac{7}{17} a^{3} - \frac{5}{34} a^{2} + \frac{8}{17} a$, $\frac{1}{102} a^{12} - \frac{1}{102} a^{11} + \frac{1}{102} a^{10} - \frac{1}{51} a^{9} - \frac{29}{102} a^{8} - \frac{1}{51} a^{7} + \frac{35}{102} a^{6} - \frac{2}{17} a^{5} + \frac{41}{102} a^{4} + \frac{9}{34} a^{3} - \frac{5}{102} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{102} a^{13} - \frac{1}{102} a^{10} + \frac{10}{51} a^{9} + \frac{10}{51} a^{8} + \frac{11}{34} a^{7} - \frac{14}{51} a^{6} - \frac{11}{51} a^{5} + \frac{1}{6} a^{4} + \frac{11}{51} a^{3} + \frac{2}{17} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{3468} a^{14} + \frac{7}{3468} a^{13} + \frac{1}{289} a^{12} + \frac{25}{1734} a^{11} + \frac{25}{3468} a^{10} - \frac{179}{867} a^{9} + \frac{671}{3468} a^{8} + \frac{1175}{3468} a^{7} - \frac{275}{3468} a^{6} + \frac{1219}{3468} a^{5} - \frac{267}{1156} a^{4} - \frac{404}{867} a^{3} - \frac{1073}{3468} a^{2} - \frac{86}{289} a + \frac{13}{51}$, $\frac{1}{922542905854476364693255557678216814621703368928925598584} a^{15} + \frac{36477412733004062188639963559166039834008835757572893}{307514301951492121564418519226072271540567789642975199528} a^{14} - \frac{619909404682291457222901102121178384563947674235157193}{461271452927238182346627778839108407310851684464462799292} a^{13} - \frac{6737843288756118010722838443138622297140471544319297}{2938034732020625365265145088147187307712431111238616556} a^{12} - \frac{5673413011678245709826891157946139511913371334763975227}{922542905854476364693255557678216814621703368928925598584} a^{11} - \frac{3493819949045829728415880709476941254254226769282777}{38439287743936515195552314903259033942570973705371899941} a^{10} - \frac{40271956985986831683022073461146081463342170408861580841}{922542905854476364693255557678216814621703368928925598584} a^{9} - \frac{18404776729864982276873160059880702855106504155810433557}{307514301951492121564418519226072271540567789642975199528} a^{8} - \frac{31878940034038607915254480348068808828934771095368553975}{922542905854476364693255557678216814621703368928925598584} a^{7} + \frac{103207325819092436395489333190200358284223983555235888837}{307514301951492121564418519226072271540567789642975199528} a^{6} - \frac{211430483854461794041008936842925363882510232115088461677}{922542905854476364693255557678216814621703368928925598584} a^{5} - \frac{4573281289811902054641779863553534170022464588836539221}{115317863231809545586656944709777101827712921116115699823} a^{4} + \frac{69092448870955845052222169003657527452707658851193758519}{307514301951492121564418519226072271540567789642975199528} a^{3} - \frac{41844960443484641797477612549904431411876237587200984773}{230635726463619091173313889419554203655425842232231399646} a^{2} - \frac{40949515602633531122478557923043725110433668204671785543}{230635726463619091173313889419554203655425842232231399646} a + \frac{5020776327676279935227767631881222698857258516437555}{43206393117950373018605074825693930995771045753509067}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 60692462322400 \) (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

\(\Q(\sqrt{1313}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{13}, \sqrt{101})\), 4.4.17069.1 x2, 4.4.132613.1 x2, 4.4.2263571297.2, 4.4.2263571297.1, 8.8.2972069112961.1, 8.8.5123755016602262209.2, 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 ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$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}$