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Properties

Label 16.8.75853444258...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $47.86$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1321, -41292, -22752, 31296, 49932, 37508, 21828, 20768, 11965, 2188, 56, -344, -198, -24, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 24*x^13 - 198*x^12 - 344*x^11 + 56*x^10 + 2188*x^9 + 11965*x^8 + 20768*x^7 + 21828*x^6 + 37508*x^5 + 49932*x^4 + 31296*x^3 - 22752*x^2 - 41292*x + 1321)
 
gp: K = bnfinit(x^16 - 12*x^14 - 24*x^13 - 198*x^12 - 344*x^11 + 56*x^10 + 2188*x^9 + 11965*x^8 + 20768*x^7 + 21828*x^6 + 37508*x^5 + 49932*x^4 + 31296*x^3 - 22752*x^2 - 41292*x + 1321, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} - 24 x^{13} - 198 x^{12} - 344 x^{11} + 56 x^{10} + 2188 x^{9} + 11965 x^{8} + 20768 x^{7} + 21828 x^{6} + 37508 x^{5} + 49932 x^{4} + 31296 x^{3} - 22752 x^{2} - 41292 x + 1321 \)

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:  \(758534442582016000000000000=2^{40}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.86$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{779} a^{12} + \frac{9}{41} a^{11} - \frac{253}{779} a^{10} - \frac{8}{41} a^{9} + \frac{290}{779} a^{8} + \frac{158}{779} a^{7} + \frac{305}{779} a^{6} + \frac{64}{779} a^{5} + \frac{314}{779} a^{4} - \frac{1}{41} a^{3} - \frac{163}{779} a^{2} - \frac{213}{779} a - \frac{374}{779}$, $\frac{1}{779} a^{13} + \frac{108}{779} a^{11} + \frac{14}{41} a^{10} - \frac{204}{779} a^{9} - \frac{355}{779} a^{8} - \frac{227}{779} a^{7} + \frac{102}{779} a^{6} + \frac{276}{779} a^{5} + \frac{2}{41} a^{4} - \frac{30}{779} a^{3} - \frac{384}{779} a^{2} + \frac{215}{779} a + \frac{4}{41}$, $\frac{1}{31939} a^{14} - \frac{17}{31939} a^{13} + \frac{10}{31939} a^{12} - \frac{7422}{31939} a^{11} + \frac{593}{31939} a^{10} - \frac{2245}{31939} a^{9} + \frac{9327}{31939} a^{8} + \frac{941}{31939} a^{7} + \frac{15392}{31939} a^{6} + \frac{5433}{31939} a^{5} - \frac{9636}{31939} a^{4} + \frac{9778}{31939} a^{3} + \frac{15706}{31939} a^{2} - \frac{2180}{31939} a - \frac{3590}{31939}$, $\frac{1}{236221890976766870889897002801} a^{15} - \frac{119231161061435222082619}{236221890976766870889897002801} a^{14} - \frac{56421090993591081002687061}{236221890976766870889897002801} a^{13} - \frac{116180531764654893836567865}{236221890976766870889897002801} a^{12} + \frac{60986717560123664652867842378}{236221890976766870889897002801} a^{11} - \frac{110855715625090605935707662546}{236221890976766870889897002801} a^{10} + \frac{22920288201020171901223939809}{236221890976766870889897002801} a^{9} + \frac{72869678403962599664829880445}{236221890976766870889897002801} a^{8} - \frac{115974377597462328290248052406}{236221890976766870889897002801} a^{7} + \frac{102813630600335591226102943747}{236221890976766870889897002801} a^{6} - \frac{485064087520892310390330224}{236221890976766870889897002801} a^{5} - \frac{38706635711841298329317485503}{236221890976766870889897002801} a^{4} + \frac{5219302187907002258261221000}{12432731104040361625784052779} a^{3} + \frac{23542820290368938283399784705}{236221890976766870889897002801} a^{2} - \frac{64000865858445059150759843802}{236221890976766870889897002801} a + \frac{8010886020196166642485699108}{236221890976766870889897002801}$

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:  \( 15373022.4643 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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}$
41Data not computed