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Properties

Label 16.0.91467766672...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 89^{8}$
Root discriminant $74.57$
Ramified primes $5, 29, 89$
Class number $10768$ (GRH)
Class group $[2, 2, 2692]$ (GRH)
Galois group $C_2^2.C_2^2:D_4$ (as 16T225)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20355859, -23181663, 21559104, -15875277, 9705318, -4470224, 2256940, -825737, 322152, -97972, 33080, -7204, 2147, -319, 79, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 79*x^14 - 319*x^13 + 2147*x^12 - 7204*x^11 + 33080*x^10 - 97972*x^9 + 322152*x^8 - 825737*x^7 + 2256940*x^6 - 4470224*x^5 + 9705318*x^4 - 15875277*x^3 + 21559104*x^2 - 23181663*x + 20355859)
 
gp: K = bnfinit(x^16 - 5*x^15 + 79*x^14 - 319*x^13 + 2147*x^12 - 7204*x^11 + 33080*x^10 - 97972*x^9 + 322152*x^8 - 825737*x^7 + 2256940*x^6 - 4470224*x^5 + 9705318*x^4 - 15875277*x^3 + 21559104*x^2 - 23181663*x + 20355859, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 79 x^{14} - 319 x^{13} + 2147 x^{12} - 7204 x^{11} + 33080 x^{10} - 97972 x^{9} + 322152 x^{8} - 825737 x^{7} + 2256940 x^{6} - 4470224 x^{5} + 9705318 x^{4} - 15875277 x^{3} + 21559104 x^{2} - 23181663 x + 20355859 \)

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:  \(914677666726224826965234765625=5^{8}\cdot 29^{6}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.57$
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:  $5, 29, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{762722351122712977991532938132029529887320621377252177} a^{15} + \frac{172900649730767425166086691097258599862276263910444842}{762722351122712977991532938132029529887320621377252177} a^{14} + \frac{294734415635724941877906978776560189613416664715700742}{762722351122712977991532938132029529887320621377252177} a^{13} - \frac{209245467940319021802127903623786303437782504431783829}{762722351122712977991532938132029529887320621377252177} a^{12} - \frac{228672843751555757232612106788503022826931206943066033}{762722351122712977991532938132029529887320621377252177} a^{11} + \frac{128476405941054032318782918063256902355195698179219457}{762722351122712977991532938132029529887320621377252177} a^{10} - \frac{335239952023726218782993428831729871572584335049852265}{762722351122712977991532938132029529887320621377252177} a^{9} - \frac{220549985086486041267290329438731513183519250739905218}{762722351122712977991532938132029529887320621377252177} a^{8} - \frac{216869766613839979384541262096911140805103578265304538}{762722351122712977991532938132029529887320621377252177} a^{7} - \frac{104073783108932381891448491113960361043321847068187581}{762722351122712977991532938132029529887320621377252177} a^{6} + \frac{75107416332333739384587333666988614806843132529484336}{762722351122712977991532938132029529887320621377252177} a^{5} + \frac{843203700947086245125220092355863682939397441388604}{762722351122712977991532938132029529887320621377252177} a^{4} + \frac{70220279773326290771050284848560704856797157650220388}{762722351122712977991532938132029529887320621377252177} a^{3} + \frac{1842900966622379444495966665254541500972183048237684}{762722351122712977991532938132029529887320621377252177} a^{2} + \frac{192846114095981425356786022207845714178918713169701490}{762722351122712977991532938132029529887320621377252177} a - \frac{16694785326835574420008245928804818395992264132641804}{40143281638037525157449102006948922625648453756697483}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2692}$, which has order $10768$ (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:  \( -1 \) (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:  \( 13998.0176198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^2:D_4$ (as 16T225):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^2.C_2^2:D_4$
Character table for $C_2^2.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.1, 4.4.725.1, 4.4.2225.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$89$89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$