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Properties

Label 16.4.15712755557...5489.7
Degree $16$
Signature $[4, 6]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20054449, 6145284, -8010384, -6068974, -750133, 1458350, 77866, -396964, 51900, 87776, 11452, -8082, -3068, -92, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 - 92*x^13 - 3068*x^12 - 8082*x^11 + 11452*x^10 + 87776*x^9 + 51900*x^8 - 396964*x^7 + 77866*x^6 + 1458350*x^5 - 750133*x^4 - 6068974*x^3 - 8010384*x^2 + 6145284*x + 20054449)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 - 92*x^13 - 3068*x^12 - 8082*x^11 + 11452*x^10 + 87776*x^9 + 51900*x^8 - 396964*x^7 + 77866*x^6 + 1458350*x^5 - 750133*x^4 - 6068974*x^3 - 8010384*x^2 + 6145284*x + 20054449, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} - 92 x^{13} - 3068 x^{12} - 8082 x^{11} + 11452 x^{10} + 87776 x^{9} + 51900 x^{8} - 396964 x^{7} + 77866 x^{6} + 1458350 x^{5} - 750133 x^{4} - 6068974 x^{3} - 8010384 x^{2} + 6145284 x + 20054449 \)

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:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{8} a$, $\frac{1}{277016} a^{14} + \frac{1842}{34627} a^{13} - \frac{14889}{277016} a^{12} - \frac{22747}{277016} a^{11} + \frac{21995}{277016} a^{10} - \frac{20947}{277016} a^{9} - \frac{10913}{138508} a^{8} - \frac{19167}{138508} a^{7} + \frac{8299}{69254} a^{6} + \frac{47445}{277016} a^{5} - \frac{8252}{34627} a^{4} + \frac{16512}{34627} a^{3} + \frac{9219}{34627} a^{2} + \frac{16917}{277016} a + \frac{101529}{277016}$, $\frac{1}{185022809076649182710009161177342543574552637704} a^{15} - \frac{127594711298253753159920644885008082972237}{185022809076649182710009161177342543574552637704} a^{14} + \frac{6989169480608320779134975033323127275866581901}{185022809076649182710009161177342543574552637704} a^{13} - \frac{6400192103179311083089018959361640346397649755}{185022809076649182710009161177342543574552637704} a^{12} + \frac{1863605649252881642402941588603002368168873621}{23127851134581147838751145147167817946819079713} a^{11} + \frac{986287094467914841573625848589901448029683813}{46255702269162295677502290294335635893638159426} a^{10} + \frac{6779909456239888121381383166861911466743123459}{92511404538324591355004580588671271787276318852} a^{9} - \frac{15358490335962054763169658067172018272423984997}{185022809076649182710009161177342543574552637704} a^{8} - \frac{10710265591854132309963160149328099631585249585}{46255702269162295677502290294335635893638159426} a^{7} - \frac{13358135846378321197359665382030633265924068025}{92511404538324591355004580588671271787276318852} a^{6} + \frac{42013306279270092074780563938615862602713644773}{185022809076649182710009161177342543574552637704} a^{5} + \frac{36475270519120849624428550743123208368280858911}{185022809076649182710009161177342543574552637704} a^{4} - \frac{72653852746829652070869146051676094749326058175}{185022809076649182710009161177342543574552637704} a^{3} + \frac{17552951000727634393159028110643525215440127445}{92511404538324591355004580588671271787276318852} a^{2} - \frac{27333466041265866177652206419397933265252232379}{185022809076649182710009161177342543574552637704} a - \frac{5526249095117998752525321475351129579235074811}{23127851134581147838751145147167817946819079713}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 147377444098 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed