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Properties

Label 16.0.13344870400...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{14}\cdot 19^{4}$
Root discriminant $24.15$
Ramified primes $2, 5, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\times C_2^3.C_4$ (as 16T99)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, 9348, 25992, 19284, 10406, 10748, 9480, 2952, 939, 996, 560, -24, 6, 32, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 + 32*x^13 + 6*x^12 - 24*x^11 + 560*x^10 + 996*x^9 + 939*x^8 + 2952*x^7 + 9480*x^6 + 10748*x^5 + 10406*x^4 + 19284*x^3 + 25992*x^2 + 9348*x + 1681)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 + 32*x^13 + 6*x^12 - 24*x^11 + 560*x^10 + 996*x^9 + 939*x^8 + 2952*x^7 + 9480*x^6 + 10748*x^5 + 10406*x^4 + 19284*x^3 + 25992*x^2 + 9348*x + 1681, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} + 32 x^{13} + 6 x^{12} - 24 x^{11} + 560 x^{10} + 996 x^{9} + 939 x^{8} + 2952 x^{7} + 9480 x^{6} + 10748 x^{5} + 10406 x^{4} + 19284 x^{3} + 25992 x^{2} + 9348 x + 1681 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13344870400000000000000=2^{24}\cdot 5^{14}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.15$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{55832324} a^{14} - \frac{617885}{13958081} a^{13} + \frac{542985}{13958081} a^{12} - \frac{973703}{13958081} a^{11} + \frac{2796297}{55832324} a^{10} - \frac{1942233}{27916162} a^{9} - \frac{2106303}{55832324} a^{8} + \frac{2714876}{13958081} a^{7} - \frac{2164077}{55832324} a^{6} + \frac{12387709}{27916162} a^{5} + \frac{4372031}{55832324} a^{4} + \frac{1491658}{13958081} a^{3} + \frac{4037755}{27916162} a^{2} + \frac{8304867}{27916162} a - \frac{220555}{1361764}$, $\frac{1}{47131514952960917776004} a^{15} + \frac{305896008395827}{47131514952960917776004} a^{14} + \frac{2315600267909657161649}{23565757476480458888002} a^{13} - \frac{953039184550841666291}{47131514952960917776004} a^{12} + \frac{181434216973937041492}{11782878738240229444001} a^{11} - \frac{10222892458685649640}{11782878738240229444001} a^{10} + \frac{2503263169977291280645}{23565757476480458888002} a^{9} - \frac{1999736771552545661843}{23565757476480458888002} a^{8} - \frac{10462687544746548188547}{23565757476480458888002} a^{7} + \frac{2225116064909022916725}{11782878738240229444001} a^{6} + \frac{9559038440352988583433}{23565757476480458888002} a^{5} + \frac{3210937519126360890539}{23565757476480458888002} a^{4} - \frac{15703729451443768378183}{47131514952960917776004} a^{3} - \frac{1825149790545486512565}{47131514952960917776004} a^{2} + \frac{5914184653807470842165}{23565757476480458888002} a - \frac{13628624462469407093}{1149549145194168726244}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2367468339397634019}{23565757476480458888002} a^{15} - \frac{18038644823073958379}{47131514952960917776004} a^{14} + \frac{32117244458446422227}{47131514952960917776004} a^{13} + \frac{166814659303820946597}{47131514952960917776004} a^{12} + \frac{10732071663573230770}{11782878738240229444001} a^{11} - \frac{208225249818524945407}{47131514952960917776004} a^{10} + \frac{2741515145599175945199}{47131514952960917776004} a^{9} + \frac{1229786553555552672977}{11782878738240229444001} a^{8} + \frac{978734934494837519044}{11782878738240229444001} a^{7} + \frac{12948715944275367143149}{47131514952960917776004} a^{6} + \frac{46364196887923842690521}{47131514952960917776004} a^{5} + \frac{10386183304649914637900}{11782878738240229444001} a^{4} + \frac{19753059248371044864825}{23565757476480458888002} a^{3} + \frac{34676210844319875127845}{23565757476480458888002} a^{2} + \frac{47431103308213952015735}{23565757476480458888002} a - \frac{614626835375351826421}{1149549145194168726244} \) (order $20$)
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:  \( 79151.1938244 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^3.C_4$ (as 16T99):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2\times C_2^3.C_4$
Character table for $C_2\times C_2^3.C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), 8.4.115520000000.2, 8.4.115520000000.1, \(\Q(\zeta_{20})\)

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$