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Properties

Label 16.4.82524197056...5625.2
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 11^{10}\cdot 19^{4}$
Root discriminant $31.25$
Ramified primes $5, 11, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1181, -8726, 3512, -2220, 8945, -3254, -2514, -1275, 204, 610, -571, -2, 10, 5, -2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 2*x^14 + 5*x^13 + 10*x^12 - 2*x^11 - 571*x^10 + 610*x^9 + 204*x^8 - 1275*x^7 - 2514*x^6 - 3254*x^5 + 8945*x^4 - 2220*x^3 + 3512*x^2 - 8726*x + 1181)
 
gp: K = bnfinit(x^16 - 3*x^15 - 2*x^14 + 5*x^13 + 10*x^12 - 2*x^11 - 571*x^10 + 610*x^9 + 204*x^8 - 1275*x^7 - 2514*x^6 - 3254*x^5 + 8945*x^4 - 2220*x^3 + 3512*x^2 - 8726*x + 1181, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 2 x^{14} + 5 x^{13} + 10 x^{12} - 2 x^{11} - 571 x^{10} + 610 x^{9} + 204 x^{8} - 1275 x^{7} - 2514 x^{6} - 3254 x^{5} + 8945 x^{4} - 2220 x^{3} + 3512 x^{2} - 8726 x + 1181 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(825241970563213134765625=5^{12}\cdot 11^{10}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.25$
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:  $5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{58} a^{13} + \frac{1}{29} a^{12} - \frac{13}{29} a^{11} - \frac{9}{58} a^{10} + \frac{14}{29} a^{9} + \frac{5}{29} a^{8} + \frac{3}{58} a^{7} + \frac{7}{29} a^{6} - \frac{9}{29} a^{5} - \frac{13}{58} a^{4} + \frac{12}{29} a^{2} + \frac{3}{58} a + \frac{12}{29}$, $\frac{1}{1102} a^{14} - \frac{5}{1102} a^{13} - \frac{69}{1102} a^{12} - \frac{175}{1102} a^{11} - \frac{141}{1102} a^{10} + \frac{75}{1102} a^{9} - \frac{67}{1102} a^{8} - \frac{471}{1102} a^{7} + \frac{13}{38} a^{6} + \frac{9}{58} a^{5} + \frac{149}{1102} a^{4} + \frac{343}{1102} a^{3} + \frac{415}{1102} a^{2} + \frac{293}{1102} a + \frac{383}{1102}$, $\frac{1}{461622485368498602822666627298} a^{15} + \frac{4477088602658593491542007}{15918016736844779407678159562} a^{14} - \frac{33606793677635334855679279}{230811242684249301411333313649} a^{13} + \frac{11622872121261956262965464382}{230811242684249301411333313649} a^{12} + \frac{149925244633970078103472031649}{461622485368498602822666627298} a^{11} - \frac{8536973558236723954588576546}{230811242684249301411333313649} a^{10} + \frac{101906144697538488746758825610}{230811242684249301411333313649} a^{9} - \frac{43300736551816157762612763453}{461622485368498602822666627298} a^{8} - \frac{38050224337174805861421100031}{230811242684249301411333313649} a^{7} + \frac{109833526363718749287540988681}{230811242684249301411333313649} a^{6} + \frac{152759631313798524640813333723}{461622485368498602822666627298} a^{5} + \frac{436540514723505390338406226}{12147960141276279021649121771} a^{4} - \frac{66184791178408376924506111665}{230811242684249301411333313649} a^{3} - \frac{52395452021894944597298633361}{461622485368498602822666627298} a^{2} + \frac{85033164699843921714110088387}{230811242684249301411333313649} a + \frac{40351127816792068543287654221}{461622485368498602822666627298}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 248960.173948 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1375.1, 4.4.15125.1, 4.2.275.1, 8.4.228765625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
$11$11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$