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Properties

Label 16.0.14542935268...3777.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 59^{8}$
Root discriminant $76.77$
Ramified primes $17, 59$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![572661, -1312335, 1436059, -745842, 587698, -274658, 111463, -13835, 5710, -2939, 1296, 115, -229, 75, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 7*x^14 + 75*x^13 - 229*x^12 + 115*x^11 + 1296*x^10 - 2939*x^9 + 5710*x^8 - 13835*x^7 + 111463*x^6 - 274658*x^5 + 587698*x^4 - 745842*x^3 + 1436059*x^2 - 1312335*x + 572661)
 
gp: K = bnfinit(x^16 - 4*x^15 - 7*x^14 + 75*x^13 - 229*x^12 + 115*x^11 + 1296*x^10 - 2939*x^9 + 5710*x^8 - 13835*x^7 + 111463*x^6 - 274658*x^5 + 587698*x^4 - 745842*x^3 + 1436059*x^2 - 1312335*x + 572661, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 7 x^{14} + 75 x^{13} - 229 x^{12} + 115 x^{11} + 1296 x^{10} - 2939 x^{9} + 5710 x^{8} - 13835 x^{7} + 111463 x^{6} - 274658 x^{5} + 587698 x^{4} - 745842 x^{3} + 1436059 x^{2} - 1312335 x + 572661 \)

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:  \(1454293526857723611028217753777=17^{13}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.77$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{13} - \frac{4}{27} a^{12} - \frac{4}{27} a^{11} - \frac{2}{27} a^{10} + \frac{1}{27} a^{9} + \frac{1}{3} a^{8} - \frac{11}{27} a^{7} + \frac{2}{27} a^{6} - \frac{2}{27} a^{5} + \frac{8}{27} a^{4} - \frac{4}{9} a^{3} - \frac{4}{27} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{5751} a^{14} - \frac{32}{5751} a^{13} - \frac{100}{639} a^{12} + \frac{92}{5751} a^{11} + \frac{283}{1917} a^{10} + \frac{1538}{5751} a^{9} - \frac{191}{5751} a^{8} - \frac{95}{5751} a^{7} + \frac{1175}{5751} a^{6} + \frac{532}{5751} a^{5} - \frac{326}{5751} a^{4} - \frac{1900}{5751} a^{3} - \frac{2768}{5751} a^{2} + \frac{319}{639} a + \frac{77}{639}$, $\frac{1}{489117439238479119147069154477944968895} a^{15} - \frac{516624742345034431537958661362207}{10869276427521758203268203432843221531} a^{14} - \frac{8411496542216170993319880980374049977}{489117439238479119147069154477944968895} a^{13} - \frac{80264293890114790600012992191478139483}{489117439238479119147069154477944968895} a^{12} + \frac{75512515236930926016133898168822937604}{489117439238479119147069154477944968895} a^{11} + \frac{4549676746370463892496482695377980511}{489117439238479119147069154477944968895} a^{10} - \frac{47495731823371539749077524658396293053}{97823487847695823829413830895588993779} a^{9} - \frac{27508024310014460016211384675321545638}{163039146412826373049023051492648322965} a^{8} - \frac{223047074906386439202787943700396973091}{489117439238479119147069154477944968895} a^{7} - \frac{228674787532607132174937957741804612304}{489117439238479119147069154477944968895} a^{6} - \frac{23990572630224084363135164359759094066}{163039146412826373049023051492648322965} a^{5} + \frac{4364580403317346909441029234207750713}{97823487847695823829413830895588993779} a^{4} - \frac{215571015914364146127613911262879381732}{489117439238479119147069154477944968895} a^{3} - \frac{21556314195596156062015863018514921103}{97823487847695823829413830895588993779} a^{2} + \frac{211364592889085145990349499360882971}{54346382137608791016341017164216107655} a + \frac{14504533468590303126007889242070051899}{54346382137608791016341017164216107655}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 502064925.746 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-59}) \), 4.0.59177.1, 8.0.17204919837377.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
$17$17.8.6.4$x^{8} + 136 x^{4} + 7803$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
17.8.7.7$x^{8} + 51$$8$$1$$7$$C_8$$[\ ]_{8}$
59Data not computed