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Properties

Label 16.0.22819400690...2368.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 73^{5}$
Root discriminant $18.72$
Ramified primes $2, 3, 73$
Class number $2$
Class group $[2]$
Galois group $C_2.D_4^2.C_2$ (as 16T659)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 430, 3152, -4268, 8035, -4396, 5182, -1808, 588, -52, -158, 138, -36, 10, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 10*x^13 - 36*x^12 + 138*x^11 - 158*x^10 - 52*x^9 + 588*x^8 - 1808*x^7 + 5182*x^6 - 4396*x^5 + 8035*x^4 - 4268*x^3 + 3152*x^2 + 430*x + 25)
 
gp: K = bnfinit(x^16 - 2*x^15 + 10*x^13 - 36*x^12 + 138*x^11 - 158*x^10 - 52*x^9 + 588*x^8 - 1808*x^7 + 5182*x^6 - 4396*x^5 + 8035*x^4 - 4268*x^3 + 3152*x^2 + 430*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 10 x^{13} - 36 x^{12} + 138 x^{11} - 158 x^{10} - 52 x^{9} + 588 x^{8} - 1808 x^{7} + 5182 x^{6} - 4396 x^{5} + 8035 x^{4} - 4268 x^{3} + 3152 x^{2} + 430 x + 25 \)

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:  \(228194006908815802368=2^{24}\cdot 3^{8}\cdot 73^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.72$
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:  $2, 3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{37} a^{13} - \frac{9}{37} a^{12} + \frac{6}{37} a^{11} - \frac{2}{37} a^{10} - \frac{13}{37} a^{9} - \frac{2}{37} a^{8} + \frac{9}{37} a^{7} - \frac{12}{37} a^{6} + \frac{15}{37} a^{5} + \frac{11}{37} a^{4} - \frac{18}{37} a^{3} - \frac{6}{37} a^{2} + \frac{16}{37} a - \frac{6}{37}$, $\frac{1}{37} a^{14} - \frac{1}{37} a^{12} + \frac{15}{37} a^{11} + \frac{6}{37} a^{10} - \frac{8}{37} a^{9} - \frac{9}{37} a^{8} - \frac{5}{37} a^{7} + \frac{18}{37} a^{6} - \frac{2}{37} a^{5} + \frac{7}{37} a^{4} + \frac{17}{37} a^{3} - \frac{1}{37} a^{2} - \frac{10}{37} a - \frac{17}{37}$, $\frac{1}{589345271932504302916285} a^{15} + \frac{1291440580708930662993}{589345271932504302916285} a^{14} + \frac{639467089768629418429}{117869054386500860583257} a^{13} + \frac{50506268279330884754038}{117869054386500860583257} a^{12} - \frac{73878695519916472009301}{589345271932504302916285} a^{11} + \frac{145335039358148985898978}{589345271932504302916285} a^{10} + \frac{282553983585916311167467}{589345271932504302916285} a^{9} + \frac{129801970289678988335628}{589345271932504302916285} a^{8} + \frac{911982471239596492774}{15928250592770386565305} a^{7} + \frac{241772064610589524802732}{589345271932504302916285} a^{6} + \frac{247316009745615015852702}{589345271932504302916285} a^{5} - \frac{1958655420740468046793}{15928250592770386565305} a^{4} + \frac{31090104698199623909507}{117869054386500860583257} a^{3} - \frac{212877178484947609909863}{589345271932504302916285} a^{2} - \frac{246070413457211013952633}{589345271932504302916285} a - \frac{1002339646627985568468}{117869054386500860583257}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{120178939186945998081}{45334251687115715608945} a^{15} + \frac{233319486924489850667}{45334251687115715608945} a^{14} - \frac{4002500836403371980}{9066850337423143121789} a^{13} - \frac{227053305665035468806}{9066850337423143121789} a^{12} + \frac{4377575374348783089611}{45334251687115715608945} a^{11} - \frac{16975650531060571197003}{45334251687115715608945} a^{10} + \frac{19030455407337766541203}{45334251687115715608945} a^{9} + \frac{5068644111287571989422}{45334251687115715608945} a^{8} - \frac{1890308213575473414399}{1225250045597722043485} a^{7} + \frac{225298467451920931255123}{45334251687115715608945} a^{6} - \frac{636717210021813510740487}{45334251687115715608945} a^{5} + \frac{14020063679793258849148}{1225250045597722043485} a^{4} - \frac{199611405668846075734073}{9066850337423143121789} a^{3} + \frac{455234448247771165442073}{45334251687115715608945} a^{2} - \frac{361751253019187893418682}{45334251687115715608945} a - \frac{12370011281191904857052}{9066850337423143121789} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8789.29006692 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.D_4^2.C_2$ (as 16T659):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 25 conjugacy class representatives for $C_2.D_4^2.C_2$
Character table for $C_2.D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 8.0.1513728.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
3Data not computed
73Data not computed