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Properties

Label 16.8.22873200555...6849.1
Degree $16$
Signature $[8, 4]$
Discriminant $37^{4}\cdot 73^{14}$
Root discriminant $105.31$
Ramified primes $37, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -35712, -289456, -507536, 247876, 994380, 312141, -113581, -72564, -11007, 4771, 2372, 147, -91, -28, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 28*x^14 - 91*x^13 + 147*x^12 + 2372*x^11 + 4771*x^10 - 11007*x^9 - 72564*x^8 - 113581*x^7 + 312141*x^6 + 994380*x^5 + 247876*x^4 - 507536*x^3 - 289456*x^2 - 35712*x + 1024)
 
gp: K = bnfinit(x^16 - x^15 - 28*x^14 - 91*x^13 + 147*x^12 + 2372*x^11 + 4771*x^10 - 11007*x^9 - 72564*x^8 - 113581*x^7 + 312141*x^6 + 994380*x^5 + 247876*x^4 - 507536*x^3 - 289456*x^2 - 35712*x + 1024, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 28 x^{14} - 91 x^{13} + 147 x^{12} + 2372 x^{11} + 4771 x^{10} - 11007 x^{9} - 72564 x^{8} - 113581 x^{7} + 312141 x^{6} + 994380 x^{5} + 247876 x^{4} - 507536 x^{3} - 289456 x^{2} - 35712 x + 1024 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(228732005557745506375281661426849=37^{4}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.31$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} - \frac{1}{16} a^{4} + \frac{3}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{7}{32} a^{6} + \frac{7}{32} a^{5} + \frac{3}{32} a^{4} + \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{64} a^{11} + \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{64} a^{7} - \frac{1}{64} a^{6} + \frac{11}{64} a^{5} + \frac{7}{32} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{14} - \frac{3}{128} a^{11} - \frac{5}{128} a^{8} - \frac{1}{32} a^{7} + \frac{7}{32} a^{6} + \frac{7}{128} a^{5} + \frac{5}{32} a^{4} + \frac{13}{32} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a$, $\frac{1}{152167066695341252316432728502272} a^{15} - \frac{484048345412768341404226845}{38041766673835313079108182125568} a^{14} - \frac{639047672191395039746462417}{4755220834229414134888522765696} a^{13} + \frac{1000879377232215610669482728581}{152167066695341252316432728502272} a^{12} - \frac{168737154933636240418572597067}{38041766673835313079108182125568} a^{11} - \frac{830856639885873786772761817171}{19020883336917656539554091062784} a^{10} + \frac{8797745598850049652007709160555}{152167066695341252316432728502272} a^{9} + \frac{18239660281117356092048779647}{9510441668458828269777045531392} a^{8} - \frac{492766212152805546908845052137}{38041766673835313079108182125568} a^{7} + \frac{29479877386192044156691692735231}{152167066695341252316432728502272} a^{6} + \frac{38839930874458626352329947701}{2377610417114707067444261382848} a^{5} - \frac{8477550768952624129851867871061}{38041766673835313079108182125568} a^{4} + \frac{6533076239777012297802498679}{594402604278676766861065345712} a^{3} + \frac{4456712074651811300235828548527}{9510441668458828269777045531392} a^{2} - \frac{97402212706101592666827767113}{1188805208557353533722130691424} a + \frac{28688494403011715697491725859}{148600651069669191715266336428}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 136240858571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.4.408753745206589.1, 8.8.15123888572643793.1, 8.4.5599366372693.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$