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Properties

Label 16.0.56118626684...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 3^{12}\cdot 5^{4}\cdot 7^{4}$
Root discriminant $40.67$
Ramified primes $2, 3, 5, 7$
Class number $56$ (GRH)
Class group $[2, 2, 14]$ (GRH)
Galois group $C_2^4:C_2^2$ (as 16T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49726, -186288, 269568, -188216, 73280, -33696, 32076, -28104, 19152, -9968, 4176, -1560, 474, -112, 24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 24*x^14 - 112*x^13 + 474*x^12 - 1560*x^11 + 4176*x^10 - 9968*x^9 + 19152*x^8 - 28104*x^7 + 32076*x^6 - 33696*x^5 + 73280*x^4 - 188216*x^3 + 269568*x^2 - 186288*x + 49726)
 
gp: K = bnfinit(x^16 - 4*x^15 + 24*x^14 - 112*x^13 + 474*x^12 - 1560*x^11 + 4176*x^10 - 9968*x^9 + 19152*x^8 - 28104*x^7 + 32076*x^6 - 33696*x^5 + 73280*x^4 - 188216*x^3 + 269568*x^2 - 186288*x + 49726, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 24 x^{14} - 112 x^{13} + 474 x^{12} - 1560 x^{11} + 4176 x^{10} - 9968 x^{9} + 19152 x^{8} - 28104 x^{7} + 32076 x^{6} - 33696 x^{5} + 73280 x^{4} - 188216 x^{3} + 269568 x^{2} - 186288 x + 49726 \)

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:  \(56118626684141976944640000=2^{46}\cdot 3^{12}\cdot 5^{4}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.67$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} - \frac{1}{13} a^{12} - \frac{1}{13} a^{11} + \frac{4}{13} a^{10} - \frac{5}{13} a^{9} - \frac{5}{13} a^{8} + \frac{5}{13} a^{7} - \frac{6}{13} a^{6} + \frac{1}{13} a^{5} - \frac{2}{13} a^{4} + \frac{6}{13} a^{3} + \frac{3}{13} a^{2} - \frac{6}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{14} - \frac{2}{13} a^{12} + \frac{3}{13} a^{11} - \frac{1}{13} a^{10} + \frac{3}{13} a^{9} - \frac{1}{13} a^{7} - \frac{5}{13} a^{6} - \frac{1}{13} a^{5} + \frac{4}{13} a^{4} - \frac{4}{13} a^{3} - \frac{3}{13} a^{2} - \frac{4}{13} a + \frac{2}{13}$, $\frac{1}{20840033714216807485781985028353625} a^{15} - \frac{86244243415705285132683534144}{6821614963737089193381991825975} a^{14} + \frac{393263583207652434768010846123619}{20840033714216807485781985028353625} a^{13} - \frac{4470592320915957042894934845003241}{20840033714216807485781985028353625} a^{12} - \frac{63179915511352699051746581586729}{4168006742843361497156397005670725} a^{11} + \frac{64174960807509175613742012828752}{4168006742843361497156397005670725} a^{10} - \frac{5745904588199028360644521353927609}{20840033714216807485781985028353625} a^{9} - \frac{7569428788765334782739395595051}{69699109412096346106294264308875} a^{8} - \frac{5708015742031074741013000149515264}{20840033714216807485781985028353625} a^{7} + \frac{240775140607018037662757439907244}{4168006742843361497156397005670725} a^{6} + \frac{6868328525334243276587654848647431}{20840033714216807485781985028353625} a^{5} + \frac{10241755083969709604584611190193008}{20840033714216807485781985028353625} a^{4} + \frac{6550293362634881968817184753010202}{20840033714216807485781985028353625} a^{3} - \frac{99129373833222244277833046350373}{20840033714216807485781985028353625} a^{2} + \frac{980229326861302998431073584209486}{20840033714216807485781985028353625} a + \frac{3755207017760656166926987146681}{19278477071430904242166498638625}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{14}$, which has order $56$ (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:  \( 63774.9292225 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2$ (as 16T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^4:C_2^2$
Character table for $C_2^4:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), 4.0.13824.1 x2, 4.0.27648.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.26011238400.1, 8.0.3745618329600.19, 8.0.3057647616.7

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

Sibling fields

Degree 16 siblings: 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 R R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$