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Properties

Label 16.0.18957657923...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 5^{12}\cdot 17^{14}$
Root discriminant $585.28$
Ramified primes $2, 5, 17$
Class number $4800118912$ (GRH)
Class group $[2, 2, 34, 34, 1038088]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![383966122500, 0, 5995981260000, 0, 911335779000, 0, 52810704000, 0, 1494203950, 0, 22344800, 0, 176460, 0, 680, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 680*x^14 + 176460*x^12 + 22344800*x^10 + 1494203950*x^8 + 52810704000*x^6 + 911335779000*x^4 + 5995981260000*x^2 + 383966122500)
 
gp: K = bnfinit(x^16 + 680*x^14 + 176460*x^12 + 22344800*x^10 + 1494203950*x^8 + 52810704000*x^6 + 911335779000*x^4 + 5995981260000*x^2 + 383966122500, 1)
 

Normalized defining polynomial

\( x^{16} + 680 x^{14} + 176460 x^{12} + 22344800 x^{10} + 1494203950 x^{8} + 52810704000 x^{6} + 911335779000 x^{4} + 5995981260000 x^{2} + 383966122500 \)

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:  \(189576579237593007104201322397696000000000000=2^{62}\cdot 5^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $585.28$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2720=2^{5}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2720}(1,·)$, $\chi_{2720}(2253,·)$, $\chi_{2720}(1997,·)$, $\chi_{2720}(2449,·)$, $\chi_{2720}(89,·)$, $\chi_{2720}(93,·)$, $\chi_{2720}(933,·)$, $\chi_{2720}(2209,·)$, $\chi_{2720}(1957,·)$, $\chi_{2720}(361,·)$, $\chi_{2720}(1437,·)$, $\chi_{2720}(2481,·)$, $\chi_{2720}(53,·)$, $\chi_{2720}(489,·)$, $\chi_{2720}(761,·)$, $\chi_{2720}(117,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{15} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{15} a^{5} + \frac{1}{3} a$, $\frac{1}{45} a^{6} + \frac{1}{45} a^{4} - \frac{4}{9} a^{2}$, $\frac{1}{135} a^{7} - \frac{2}{135} a^{5} + \frac{2}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{22950} a^{8} - \frac{1}{135} a^{6} - \frac{4}{135} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{68850} a^{9} + \frac{1}{135} a^{5} - \frac{13}{81} a^{3} + \frac{4}{9} a$, $\frac{1}{1032750} a^{10} - \frac{17}{2025} a^{6} - \frac{8}{243} a^{4} - \frac{44}{135} a^{2}$, $\frac{1}{1032750} a^{11} - \frac{2}{2025} a^{7} + \frac{23}{1215} a^{5} + \frac{11}{135} a^{3} - \frac{1}{3} a$, $\frac{1}{1032750} a^{12} + \frac{1}{68850} a^{8} + \frac{1}{243} a^{6} + \frac{1}{45} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9294750} a^{13} - \frac{2}{4647375} a^{11} + \frac{2}{309825} a^{9} - \frac{131}{54675} a^{7} - \frac{58}{2187} a^{5} + \frac{91}{1215} a^{3} + \frac{11}{27} a$, $\frac{1}{185502323506440288071250} a^{14} + \frac{5983366007253461}{18550232350644028807125} a^{12} - \frac{2150634589171891}{6183410783548009602375} a^{10} - \frac{149993657376911251}{7420092940257611522850} a^{8} - \frac{73121252316648194}{218238027654635633025} a^{6} - \frac{30059903073158489}{969946789576158369} a^{4} - \frac{161255652601505458}{538859327542310205} a^{2} + \frac{154446118923003}{443505619376387}$, $\frac{1}{556506970519320864213750} a^{15} - \frac{465865215031}{6547140829639068990750} a^{13} + \frac{233298662384123}{742009294025761152285} a^{11} + \frac{14813059235262742}{11130139410386417284275} a^{9} - \frac{24203971841859598}{130942816592781379815} a^{7} + \frac{83034029867820196}{2909840368728475107} a^{5} - \frac{222459428075446864}{1616577982626930615} a^{3} + \frac{503003831554253}{11974651723162449} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{34}\times C_{34}\times C_{1038088}$, which has order $4800118912$ (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:  \( 308723659.32686925 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{10}, \sqrt{34})\), 4.4.251545600.1, 4.4.10061824.2, 8.8.63275188879360000.2, 8.0.13768681100148736000000.5, 8.0.13768681100148736000000.1

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17Data not computed