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Properties

Label 16.16.2090117913...4096.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{58}\cdot 3^{14}\cdot 11^{8}\cdot 29^{4}$
Root discriminant $248.32$
Ramified primes $2, 3, 11, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2^2\times D_4).C_2^3$ (as 16T600)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![838779390801, 0, -1472465388240, 0, 276139463688, 0, -19886429376, 0, 679356150, 0, -11929632, 0, 111432, 0, -528, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 528*x^14 + 111432*x^12 - 11929632*x^10 + 679356150*x^8 - 19886429376*x^6 + 276139463688*x^4 - 1472465388240*x^2 + 838779390801)
 
gp: K = bnfinit(x^16 - 528*x^14 + 111432*x^12 - 11929632*x^10 + 679356150*x^8 - 19886429376*x^6 + 276139463688*x^4 - 1472465388240*x^2 + 838779390801, 1)
 

Normalized defining polynomial

\( x^{16} - 528 x^{14} + 111432 x^{12} - 11929632 x^{10} + 679356150 x^{8} - 19886429376 x^{6} + 276139463688 x^{4} - 1472465388240 x^{2} + 838779390801 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(209011791362878677931936734396700164096=2^{58}\cdot 3^{14}\cdot 11^{8}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $248.32$
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:  $2, 3, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{66} a^{8} + \frac{4}{11} a^{4} - \frac{1}{2}$, $\frac{1}{66} a^{9} + \frac{4}{11} a^{5} - \frac{1}{2} a$, $\frac{1}{21054} a^{10} + \frac{5}{957} a^{8} + \frac{1735}{7018} a^{6} - \frac{251}{638} a^{4} + \frac{73}{319} a^{2} - \frac{1}{2}$, $\frac{1}{126324} a^{11} - \frac{1}{42108} a^{10} + \frac{13}{3828} a^{9} + \frac{19}{3828} a^{8} - \frac{2198}{10527} a^{7} + \frac{887}{7018} a^{6} - \frac{244}{957} a^{5} - \frac{237}{638} a^{4} - \frac{173}{3828} a^{3} - \frac{465}{1276} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{18316980} a^{12} - \frac{59}{3052830} a^{10} - \frac{2649}{407044} a^{8} + \frac{604937}{3052830} a^{6} - \frac{23569}{50460} a^{4} - \frac{59}{1595} a^{2} - \frac{1}{20}$, $\frac{1}{18316980} a^{13} - \frac{16}{4579245} a^{11} + \frac{347}{1221132} a^{9} - \frac{223301}{1017610} a^{7} + \frac{12761}{555060} a^{5} - \frac{1219}{9570} a^{3} + \frac{9}{20} a$, $\frac{1}{512244865042796241191786940} a^{14} + \frac{12592136944807996}{1293547638996960205029765} a^{12} + \frac{597225358917815878157}{34149657669519749412785796} a^{10} - \frac{16463079195065293542083}{7761285833981761230178590} a^{8} - \frac{3659685087010137953848769}{15522571667963522460357180} a^{6} - \frac{488113172749041477963}{1351669424239247863145} a^{4} + \frac{84467911341074004303}{186437161964034188020} a^{2} + \frac{23959495477088737}{58444251399383758}$, $\frac{1}{512244865042796241191786940} a^{15} + \frac{12592136944807996}{1293547638996960205029765} a^{13} + \frac{18852529740705577633}{11383219223173249804261932} a^{11} + \frac{48417053168418603888877}{7761285833981761230178590} a^{9} + \frac{2822429955487025980389871}{15522571667963522460357180} a^{7} - \frac{1977041713648218450944}{4055008272717743589435} a^{5} - \frac{255353474408413600481}{559311485892102564060} a^{3} + \frac{23959495477088737}{58444251399383758} a$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $15$
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:  \( 53926036048600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times D_4).C_2^3$ (as 16T600):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $(C_2^2\times D_4).C_2^3$
Character table for $(C_2^2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.76032.1, 4.4.13824.1, 4.4.50688.2, 8.8.369975361536.2

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$