↑

Properties

Label 16.8.30187924998...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 11^{2}\cdot 29^{6}$
Root discriminant $52.18$
Ramified primes $2, 5, 11, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T799

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5921, 133848, 222548, -35720, -325808, -382176, -231090, -78848, -6115, 8168, 4782, 1048, -44, -80, -26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 26*x^14 - 80*x^13 - 44*x^12 + 1048*x^11 + 4782*x^10 + 8168*x^9 - 6115*x^8 - 78848*x^7 - 231090*x^6 - 382176*x^5 - 325808*x^4 - 35720*x^3 + 222548*x^2 + 133848*x + 5921)
 
gp: K = bnfinit(x^16 - 26*x^14 - 80*x^13 - 44*x^12 + 1048*x^11 + 4782*x^10 + 8168*x^9 - 6115*x^8 - 78848*x^7 - 231090*x^6 - 382176*x^5 - 325808*x^4 - 35720*x^3 + 222548*x^2 + 133848*x + 5921, 1)
 

Normalized defining polynomial

\( x^{16} - 26 x^{14} - 80 x^{13} - 44 x^{12} + 1048 x^{11} + 4782 x^{10} + 8168 x^{9} - 6115 x^{8} - 78848 x^{7} - 231090 x^{6} - 382176 x^{5} - 325808 x^{4} - 35720 x^{3} + 222548 x^{2} + 133848 x + 5921 \)

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:  \(3018792499821936640000000000=2^{32}\cdot 5^{10}\cdot 11^{2}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.18$
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:  $2, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{3}{10} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{2}{5} a^{5} + \frac{3}{10} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{11} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{3}{10} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{10} a^{2} - \frac{2}{5} a - \frac{1}{10}$, $\frac{1}{2370298791335606359885826928045823610} a^{15} - \frac{31631556788600319594412249490713323}{1185149395667803179942913464022911805} a^{14} - \frac{34649594337964852093084428220581991}{1185149395667803179942913464022911805} a^{13} - \frac{52646477920521620627394581027554237}{2370298791335606359885826928045823610} a^{12} - \frac{79225068094476761087604335241311439}{1185149395667803179942913464022911805} a^{11} - \frac{80536449689394482661655900905303852}{1185149395667803179942913464022911805} a^{10} - \frac{115641525441565274523513707847856008}{1185149395667803179942913464022911805} a^{9} - \frac{192850486472448529926633193034905659}{1185149395667803179942913464022911805} a^{8} - \frac{347533610140709663765887520908635722}{1185149395667803179942913464022911805} a^{7} - \frac{293093161215061605457926653681462912}{1185149395667803179942913464022911805} a^{6} + \frac{505856210800956486329582998061993281}{1185149395667803179942913464022911805} a^{5} + \frac{42900894604284408278607836125245433}{1185149395667803179942913464022911805} a^{4} - \frac{31140086182539197136431805025222041}{2370298791335606359885826928045823610} a^{3} - \frac{585228105831267731018196181841892824}{1185149395667803179942913464022911805} a^{2} + \frac{83103180268180938687959923390496223}{237029879133560635988582692804582361} a - \frac{1241195853394854694280970504022801}{15292250266681331354102109213198862}$

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:  $11$
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:  \( 27637676.1785 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.4.46400.1, 4.4.725.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.2152960000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.2.1$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$