↑

Properties

Label 16.8.85171535827...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{6}\cdot 409^{2}\cdot 761^{6}$
Root discriminant $132.02$
Ramified primes $2, 5, 409, 761$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1127

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-62213719, -24597176, 168658082, -101529080, 43234682, -23486622, 2227616, -1272146, 1310177, -124402, 65721, 12454, -7876, 582, -85, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 85*x^14 + 582*x^13 - 7876*x^12 + 12454*x^11 + 65721*x^10 - 124402*x^9 + 1310177*x^8 - 1272146*x^7 + 2227616*x^6 - 23486622*x^5 + 43234682*x^4 - 101529080*x^3 + 168658082*x^2 - 24597176*x - 62213719)
 
gp: K = bnfinit(x^16 - 2*x^15 - 85*x^14 + 582*x^13 - 7876*x^12 + 12454*x^11 + 65721*x^10 - 124402*x^9 + 1310177*x^8 - 1272146*x^7 + 2227616*x^6 - 23486622*x^5 + 43234682*x^4 - 101529080*x^3 + 168658082*x^2 - 24597176*x - 62213719, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 85 x^{14} + 582 x^{13} - 7876 x^{12} + 12454 x^{11} + 65721 x^{10} - 124402 x^{9} + 1310177 x^{8} - 1272146 x^{7} + 2227616 x^{6} - 23486622 x^{5} + 43234682 x^{4} - 101529080 x^{3} + 168658082 x^{2} - 24597176 x - 62213719 \)

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:  \(8517153582795178291400802304000000=2^{24}\cdot 5^{6}\cdot 409^{2}\cdot 761^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.02$
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:  $2, 5, 409, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{15} + \frac{2778053301731592063645403876148601216622180998715535367139016139549623}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{14} - \frac{2422363739781709409433996610106532008474211632091883746681111025573627}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{13} - \frac{2442028516495917815501406036109746255930250624552077724235033591839128}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{12} + \frac{1874009674401685037231930752596550261617205719339535444319924348867466}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{11} - \frac{1586437333431965880363968531192319710360538384090355328266518699855142}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{10} + \frac{1924253234579215592199233215327466508086540439125650692214072673965325}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{9} - \frac{3185283939023822483613597810697751415109043539034377116437006068983357}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{8} - \frac{2696567329319426763033565806600460281567463729754824094676321728205615}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{7} + \frac{3143067253263411956742758245081552514822879200445073579177261226766792}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{6} - \frac{20261188068496615951936410516609597048442141915242880318694145425149}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{5} + \frac{1301551145935453965285705397610918457132821713133286300556753300858798}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{4} - \frac{2475974034293996421590463086509562281060175109007877777154534447752938}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{3} - \frac{2543850835680001592042318734526378254835184760824859511183682670207859}{7799189882343827208553945193592667669366464444277555453139182303262807} a^{2} - \frac{1432564850461320506293475380039213615877467156443971088339720054245512}{7799189882343827208553945193592667669366464444277555453139182303262807} a + \frac{986882514227666001715063938693927614105852145355617200170869935442915}{7799189882343827208553945193592667669366464444277555453139182303262807}$

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:  $11$
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:  \( 21361254063.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1127:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 58 conjugacy class representatives for t16n1127 are not computed
Character table for t16n1127 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.48704.1, 8.8.59301990400.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
409Data not computed
761Data not computed