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Properties

Label 16.0.22802876945...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{10}\cdot 19^{6}\cdot 59^{4}$
Root discriminant $38.45$
Ramified primes $2, 5, 19, 59$
Class number $12$
Class group $[2, 6]$
Galois group 16T1174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10000, 0, 37000, 0, 37745, 0, 12615, 0, 2366, 0, 343, 0, 18, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 + 18*x^12 + 343*x^10 + 2366*x^8 + 12615*x^6 + 37745*x^4 + 37000*x^2 + 10000)
 
gp: K = bnfinit(x^16 + 3*x^14 + 18*x^12 + 343*x^10 + 2366*x^8 + 12615*x^6 + 37745*x^4 + 37000*x^2 + 10000, 1)
 

Normalized defining polynomial

\( x^{16} + 3 x^{14} + 18 x^{12} + 343 x^{10} + 2366 x^{8} + 12615 x^{6} + 37745 x^{4} + 37000 x^{2} + 10000 \)

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:  \(22802876945601640000000000=2^{12}\cdot 5^{10}\cdot 19^{6}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.45$
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:  $2, 5, 19, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{80} a^{12} - \frac{1}{16} a^{11} + \frac{3}{80} a^{10} - \frac{1}{8} a^{9} + \frac{3}{80} a^{8} - \frac{1}{16} a^{7} + \frac{13}{80} a^{6} - \frac{1}{4} a^{5} - \frac{9}{80} a^{4} + \frac{5}{16} a^{3} - \frac{5}{16} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{80} a^{13} - \frac{1}{40} a^{11} - \frac{1}{16} a^{10} - \frac{7}{80} a^{9} - \frac{1}{8} a^{8} + \frac{1}{10} a^{7} - \frac{1}{16} a^{6} + \frac{11}{80} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{5}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{348343559200} a^{14} - \frac{721914607}{348343559200} a^{12} - \frac{1}{16} a^{11} - \frac{1977306589}{43542944900} a^{10} - \frac{1}{8} a^{9} - \frac{21257321187}{348343559200} a^{8} - \frac{1}{16} a^{7} - \frac{570425029}{21771472450} a^{6} - \frac{1764802577}{9952673120} a^{4} - \frac{7}{16} a^{3} - \frac{2746737781}{69668711840} a^{2} - \frac{1}{2} a + \frac{542287729}{3483435592}$, $\frac{1}{6966871184000} a^{15} - \frac{1}{696687118400} a^{14} - \frac{5076209097}{6966871184000} a^{13} - \frac{3632379883}{696687118400} a^{12} - \frac{210383920141}{3483435592000} a^{11} - \frac{20393687829}{348343559200} a^{10} + \frac{314023354543}{6966871184000} a^{9} + \frac{8194437717}{696687118400} a^{8} + \frac{424334607033}{3483435592000} a^{7} - \frac{1968041503}{348343559200} a^{6} - \frac{47920324171}{199053462400} a^{5} - \frac{9556363097}{19905346240} a^{4} + \frac{201905103249}{1393374236800} a^{3} - \frac{36441912629}{139337423680} a^{2} + \frac{1327660003}{13933742368} a + \frac{2070288965}{6966871184}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 490635.029761 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 70 conjugacy class representatives for t16n1174 are not computed
Character table for t16n1174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.28025.1, 8.4.1193808950000.1, 8.0.62832050000.1, 8.4.238761790000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$