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Properties

Label 16.0.80661021875...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 5^{12}\cdot 17^{15}$
Root discriminant $554.84$
Ramified primes $2, 3, 5, 17$
Class number $1197694016$ (GRH)
Class group $[2, 2, 4, 74855876]$ (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![548093933370000, 0, 107755895700000, 0, 8264854521000, 0, 316195002000, 0, 6441606000, 0, 69859800, 0, 387090, 0, 1020, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1020*x^14 + 387090*x^12 + 69859800*x^10 + 6441606000*x^8 + 316195002000*x^6 + 8264854521000*x^4 + 107755895700000*x^2 + 548093933370000)
 
gp: K = bnfinit(x^16 + 1020*x^14 + 387090*x^12 + 69859800*x^10 + 6441606000*x^8 + 316195002000*x^6 + 8264854521000*x^4 + 107755895700000*x^2 + 548093933370000, 1)
 

Normalized defining polynomial

\( x^{16} + 1020 x^{14} + 387090 x^{12} + 69859800 x^{10} + 6441606000 x^{8} + 316195002000 x^{6} + 8264854521000 x^{4} + 107755895700000 x^{2} + 548093933370000 \)

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:  \(80661021875089306767964564881408000000000000=2^{44}\cdot 3^{8}\cdot 5^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $554.84$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(2693,·)$, $\chi_{4080}(3841,·)$, $\chi_{4080}(2569,·)$, $\chi_{4080}(3437,·)$, $\chi_{4080}(1441,·)$, $\chi_{4080}(533,·)$, $\chi_{4080}(3289,·)$, $\chi_{4080}(1369,·)$, $\chi_{4080}(3677,·)$, $\chi_{4080}(2401,·)$, $\chi_{4080}(3173,·)$, $\chi_{4080}(2089,·)$, $\chi_{4080}(2477,·)$, $\chi_{4080}(2717,·)$, $\chi_{4080}(1013,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{90} a^{4}$, $\frac{1}{90} a^{5}$, $\frac{1}{270} a^{6}$, $\frac{1}{270} a^{7}$, $\frac{1}{8100} a^{8}$, $\frac{1}{8100} a^{9}$, $\frac{1}{24300} a^{10}$, $\frac{1}{24300} a^{11}$, $\frac{1}{2916000} a^{12} - \frac{1}{540} a^{6} + \frac{1}{4}$, $\frac{1}{2044116000} a^{13} - \frac{41}{4258575} a^{11} - \frac{79}{1419525} a^{9} + \frac{377}{378540} a^{7} + \frac{137}{63090} a^{5} - \frac{206}{2103} a^{3} - \frac{235}{2804} a$, $\frac{1}{5035963317852750482258172000} a^{14} - \frac{135401854806725124469}{1678654439284250160752724000} a^{12} + \frac{8463315421635941929}{2331464499005903001045450} a^{10} - \frac{237926894080364072153}{4662928998011806002090900} a^{8} - \frac{343777555128330701387}{310861933200787066806060} a^{6} - \frac{14225716450043827667}{51810322200131177801010} a^{4} + \frac{605102907765990352285}{6908042960017490373468} a^{2} + \frac{119904527137563607}{3284851621501421956}$, $\frac{1}{5035963317852750482258172000} a^{15} + \frac{6142161263033201}{104915902455265635047045250} a^{13} + \frac{18659886208635988867}{932585799602361200418180} a^{11} + \frac{59650359878957782781}{2331464499005903001045450} a^{9} + \frac{40394758475364043213}{155430966600393533403030} a^{7} + \frac{4261470539329755731}{1727010740004372593367} a^{5} - \frac{518316346787495956667}{6908042960017490373468} a^{3} + \frac{119764120252244519679}{575670246668124197789} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{74855876}$, which has order $1197694016$ (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:  \( 2470951.49590257 \) (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, 8.8.1050467002880000.1

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 R R R $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ 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
$2$2.8.22.31$x^{8} + 12 x^{6} + 6 x^{4} + 8 x^{2} + 52$$4$$2$$22$$C_8$$[3, 4]^{2}$
2.8.22.31$x^{8} + 12 x^{6} + 6 x^{4} + 8 x^{2} + 52$$4$$2$$22$$C_8$$[3, 4]^{2}$
3Data not computed
5Data not computed
17Data not computed