↑

Properties

Label 16.0.11544476107...0569.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 71^{6}$
Root discriminant $116.52$
Ramified primes $37, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20958289, -29709493, -19828694, 14111857, 15767727, 4996370, 1269996, 739018, 323371, 72254, 5679, 1324, 718, 158, 18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 18*x^14 + 158*x^13 + 718*x^12 + 1324*x^11 + 5679*x^10 + 72254*x^9 + 323371*x^8 + 739018*x^7 + 1269996*x^6 + 4996370*x^5 + 15767727*x^4 + 14111857*x^3 - 19828694*x^2 - 29709493*x + 20958289)
 
gp: K = bnfinit(x^16 - 2*x^15 + 18*x^14 + 158*x^13 + 718*x^12 + 1324*x^11 + 5679*x^10 + 72254*x^9 + 323371*x^8 + 739018*x^7 + 1269996*x^6 + 4996370*x^5 + 15767727*x^4 + 14111857*x^3 - 19828694*x^2 - 29709493*x + 20958289, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 18 x^{14} + 158 x^{13} + 718 x^{12} + 1324 x^{11} + 5679 x^{10} + 72254 x^{9} + 323371 x^{8} + 739018 x^{7} + 1269996 x^{6} + 4996370 x^{5} + 15767727 x^{4} + 14111857 x^{3} - 19828694 x^{2} - 29709493 x + 20958289 \)

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:  \(1154447610730415783388912356000569=37^{14}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.52$
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:  $37, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{4}{11} a^{12} + \frac{3}{11} a^{11} - \frac{2}{11} a^{10} - \frac{5}{11} a^{9} - \frac{3}{11} a^{8} + \frac{4}{11} a^{7} + \frac{4}{11} a^{6} + \frac{4}{11} a^{5} + \frac{5}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a$, $\frac{1}{64823} a^{14} + \frac{681}{64823} a^{13} + \frac{5571}{64823} a^{12} + \frac{31905}{64823} a^{11} + \frac{13722}{64823} a^{10} - \frac{328}{5893} a^{9} - \frac{15370}{64823} a^{8} + \frac{26197}{64823} a^{7} - \frac{5747}{64823} a^{6} - \frac{30969}{64823} a^{5} - \frac{27836}{64823} a^{4} + \frac{18925}{64823} a^{3} + \frac{13568}{64823} a^{2} + \frac{17539}{64823} a - \frac{864}{5893}$, $\frac{1}{29327633256007008850234803024461573141650712646281} a^{15} - \frac{29713585293964284219153518053909195983099512}{29327633256007008850234803024461573141650712646281} a^{14} + \frac{443171123977320947588814920860248007863734140987}{29327633256007008850234803024461573141650712646281} a^{13} + \frac{9866354813495040425444462865279918408606473940436}{29327633256007008850234803024461573141650712646281} a^{12} - \frac{12428017268480947586347444223991816929395022292997}{29327633256007008850234803024461573141650712646281} a^{11} - \frac{3866714166700548464115511421952822895996949682169}{29327633256007008850234803024461573141650712646281} a^{10} + \frac{10495379732285330530449384453083189579736467743916}{29327633256007008850234803024461573141650712646281} a^{9} - \frac{11071525040889965287533730102980662895391308252515}{29327633256007008850234803024461573141650712646281} a^{8} - \frac{2700997471287844242639669522452149144686336956866}{29327633256007008850234803024461573141650712646281} a^{7} - \frac{4426384365604878855495226883907923028546159093788}{29327633256007008850234803024461573141650712646281} a^{6} - \frac{11284640190554870713804960697078511936412510458483}{29327633256007008850234803024461573141650712646281} a^{5} - \frac{2181473157212146843149855728136673855692793628888}{29327633256007008850234803024461573141650712646281} a^{4} + \frac{4330293332440637082528040844539347898341487553132}{29327633256007008850234803024461573141650712646281} a^{3} + \frac{4385913841142633961738616557603977899057821880}{715308128195292898786214707913696905893919820641} a^{2} + \frac{7994407578056719288549326619648157080941841107233}{29327633256007008850234803024461573141650712646281} a + \frac{1316101947291099607037277024065138323085131160767}{2666148477818818986384982093132870285604610240571}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 13120388619.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 4.2.3596363.1, 4.2.97199.1, 8.0.478551592627453.1 x2, 8.0.12933826827769.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
71Data not computed