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Properties

Label 16.0.71744521380...1481.3
Degree $16$
Signature $[0, 8]$
Discriminant $23^{12}\cdot 41^{9}$
Root discriminant $84.82$
Ramified primes $23, 41$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21924773, 120480118, 166889104, 2748753, 22518941, -4737977, 2681016, -555097, 212754, -41501, 7906, -1690, 374, 51, -14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 14*x^14 + 51*x^13 + 374*x^12 - 1690*x^11 + 7906*x^10 - 41501*x^9 + 212754*x^8 - 555097*x^7 + 2681016*x^6 - 4737977*x^5 + 22518941*x^4 + 2748753*x^3 + 166889104*x^2 + 120480118*x + 21924773)
 
gp: K = bnfinit(x^16 - x^15 - 14*x^14 + 51*x^13 + 374*x^12 - 1690*x^11 + 7906*x^10 - 41501*x^9 + 212754*x^8 - 555097*x^7 + 2681016*x^6 - 4737977*x^5 + 22518941*x^4 + 2748753*x^3 + 166889104*x^2 + 120480118*x + 21924773, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 14 x^{14} + 51 x^{13} + 374 x^{12} - 1690 x^{11} + 7906 x^{10} - 41501 x^{9} + 212754 x^{8} - 555097 x^{7} + 2681016 x^{6} - 4737977 x^{5} + 22518941 x^{4} + 2748753 x^{3} + 166889104 x^{2} + 120480118 x + 21924773 \)

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:  \(7174452138071971572143971681481=23^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.82$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{244164801020104952404387566212394405763785030139517235504888391} a^{15} - \frac{36312539870870912057931853138739646788206326875451370440092116}{244164801020104952404387566212394405763785030139517235504888391} a^{14} + \frac{1942114346438364905920633355926662784090881205736231175504889}{8419475897244998358771985041806703647027070004810939155340979} a^{13} + \frac{111293101637550361468533651827034870513645344132905808681448177}{244164801020104952404387566212394405763785030139517235504888391} a^{12} - \frac{93546665043407914876132595162621234561393639046115022577367305}{244164801020104952404387566212394405763785030139517235504888391} a^{11} - \frac{35009035672866499937683228105527693878893193363769832396435940}{244164801020104952404387566212394405763785030139517235504888391} a^{10} + \frac{60326538109969312628027610657330309597876468963160220566791441}{244164801020104952404387566212394405763785030139517235504888391} a^{9} + \frac{77105913360133356307944743556004542584377300969607489836215546}{244164801020104952404387566212394405763785030139517235504888391} a^{8} - \frac{103894469634360621403315812056566133573069190415377161346987263}{244164801020104952404387566212394405763785030139517235504888391} a^{7} + \frac{58535853838143449637051632790796818274646496901704988815130423}{244164801020104952404387566212394405763785030139517235504888391} a^{6} - \frac{20029726095857259832722346251737217771747209973967317804112653}{244164801020104952404387566212394405763785030139517235504888391} a^{5} + \frac{49792687136919173856775568477713243497961174386368733276994958}{244164801020104952404387566212394405763785030139517235504888391} a^{4} - \frac{31615524033498352760392807507990179889975419478982736453869863}{244164801020104952404387566212394405763785030139517235504888391} a^{3} + \frac{88302916899457955154577020516620717044594288082654056819127510}{244164801020104952404387566212394405763785030139517235504888391} a^{2} + \frac{27930312382640255349486292946320581910648059126400588680556173}{244164801020104952404387566212394405763785030139517235504888391} a - \frac{94311859372978290704517830266938116058673275968685507710254276}{244164801020104952404387566212394405763785030139517235504888391}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  \( 26099939.1227 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), 4.0.21689.1, 8.0.19286921561.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $16$ $16$ R ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$23$23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed