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Properties

Label 16.0.45850482521...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 17^{15}$
Root discriminant $82.48$
Ramified primes $3, 5, 17$
Class number $15112$ (GRH)
Class group $[2, 7556]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![459511, -2023749, 2686052, -901018, 129932, 420375, 217448, 76584, 67695, 4776, 10728, -375, 1072, -52, 52, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 52*x^14 - 52*x^13 + 1072*x^12 - 375*x^11 + 10728*x^10 + 4776*x^9 + 67695*x^8 + 76584*x^7 + 217448*x^6 + 420375*x^5 + 129932*x^4 - 901018*x^3 + 2686052*x^2 - 2023749*x + 459511)
 
gp: K = bnfinit(x^16 - x^15 + 52*x^14 - 52*x^13 + 1072*x^12 - 375*x^11 + 10728*x^10 + 4776*x^9 + 67695*x^8 + 76584*x^7 + 217448*x^6 + 420375*x^5 + 129932*x^4 - 901018*x^3 + 2686052*x^2 - 2023749*x + 459511, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 52 x^{14} - 52 x^{13} + 1072 x^{12} - 375 x^{11} + 10728 x^{10} + 4776 x^{9} + 67695 x^{8} + 76584 x^{7} + 217448 x^{6} + 420375 x^{5} + 129932 x^{4} - 901018 x^{3} + 2686052 x^{2} - 2023749 x + 459511 \)

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:  \(4585048252186499369597900390625=3^{8}\cdot 5^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.48$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(49,·)$, $\chi_{255}(143,·)$, $\chi_{255}(16,·)$, $\chi_{255}(19,·)$, $\chi_{255}(23,·)$, $\chi_{255}(94,·)$, $\chi_{255}(229,·)$, $\chi_{255}(166,·)$, $\chi_{255}(167,·)$, $\chi_{255}(106,·)$, $\chi_{255}(107,·)$, $\chi_{255}(113,·)$, $\chi_{255}(182,·)$, $\chi_{255}(248,·)$, $\chi_{255}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{3}{13} a^{9} + \frac{4}{13} a^{8} - \frac{6}{13} a^{7} - \frac{1}{13} a^{6} - \frac{3}{13} a^{5} + \frac{3}{13} a^{4} + \frac{2}{13} a^{3} + \frac{6}{13} a^{2} - \frac{3}{13} a$, $\frac{1}{13} a^{11} - \frac{5}{13} a^{9} + \frac{6}{13} a^{8} - \frac{6}{13} a^{7} - \frac{6}{13} a^{6} - \frac{6}{13} a^{5} - \frac{2}{13} a^{4} - \frac{1}{13} a^{3} + \frac{2}{13} a^{2} + \frac{4}{13} a$, $\frac{1}{13} a^{12} + \frac{4}{13} a^{9} + \frac{1}{13} a^{8} + \frac{3}{13} a^{7} + \frac{2}{13} a^{6} - \frac{4}{13} a^{5} + \frac{1}{13} a^{4} - \frac{1}{13} a^{3} - \frac{5}{13} a^{2} - \frac{2}{13} a$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{85214852473206147226880413976616708361224083} a^{15} + \frac{1174757393397548099591719089484279153301210}{85214852473206147226880413976616708361224083} a^{14} + \frac{2310752873149306579197053087795924268814914}{85214852473206147226880413976616708361224083} a^{13} - \frac{3133909258645420602763367747842229409291775}{85214852473206147226880413976616708361224083} a^{12} + \frac{2008501783178689507769060346176460401491864}{85214852473206147226880413976616708361224083} a^{11} + \frac{2551241921293177174447404639633755558681}{139467843655001877621735538423267935124753} a^{10} + \frac{11915369844168838100774580344895468052703828}{85214852473206147226880413976616708361224083} a^{9} + \frac{12874059568394986599713493071803735727123825}{85214852473206147226880413976616708361224083} a^{8} + \frac{24513591633466314611743190413540019688706708}{85214852473206147226880413976616708361224083} a^{7} - \frac{36609363077934221052955972267602646014076329}{85214852473206147226880413976616708361224083} a^{6} - \frac{1820254136266575383963133407919656525565929}{6554988651785088248221570305893592950863391} a^{5} + \frac{5220191686481242634146570042159495815479173}{85214852473206147226880413976616708361224083} a^{4} + \frac{34086240457797456623833763837781550330996328}{85214852473206147226880413976616708361224083} a^{3} - \frac{1460533343586668557885326550136025888044418}{6554988651785088248221570305893592950863391} a^{2} - \frac{25836493387116187758059806583783123016112649}{85214852473206147226880413976616708361224083} a + \frac{2733690665737037974366297871686082453316334}{6554988651785088248221570305893592950863391}$

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

Class group and class number

$C_{2}\times C_{7556}$, which has order $15112$ (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:  \( 81485.0410293661 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

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

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.256461670625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R $16$ $16$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
5Data not computed
17Data not computed