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Properties

Label 16.0.10771233348...8801.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 73^{4}$
Root discriminant $75.34$
Ramified primes $41, 73$
Class number $33$ (GRH)
Class group $[33]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5607424, -5948416, 4678656, -2860288, 2015984, -1117344, 576600, -227294, 85127, -26439, 8234, -1761, 425, -53, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424)
 
gp: K = bnfinit(x^16 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 20 x^{14} - 53 x^{13} + 425 x^{12} - 1761 x^{11} + 8234 x^{10} - 26439 x^{9} + 85127 x^{8} - 227294 x^{7} + 576600 x^{6} - 1117344 x^{5} + 2015984 x^{4} - 2860288 x^{3} + 4678656 x^{2} - 5948416 x + 5607424 \)

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:  \(1077123334824966013513439398801=41^{14}\cdot 73^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.34$
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:  $41, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{3}{8} a^{8} - \frac{3}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{3}{16} a^{9} - \frac{3}{16} a^{8} + \frac{3}{16} a^{7} - \frac{1}{8} a^{6} - \frac{7}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{5}{64} a^{10} - \frac{11}{64} a^{9} + \frac{3}{64} a^{8} + \frac{3}{32} a^{7} + \frac{1}{64} a^{6} + \frac{11}{64} a^{5} - \frac{1}{32} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{468224} a^{14} - \frac{673}{468224} a^{13} - \frac{167}{14632} a^{12} + \frac{197}{15104} a^{11} + \frac{17077}{468224} a^{10} + \frac{24099}{468224} a^{9} + \frac{15491}{234112} a^{8} - \frac{199583}{468224} a^{7} + \frac{96395}{468224} a^{6} + \frac{116911}{234112} a^{5} + \frac{6463}{14632} a^{4} - \frac{655}{3658} a^{3} - \frac{5793}{29264} a^{2} + \frac{2559}{7316} a + \frac{564}{1829}$, $\frac{1}{148742268715640063595016247420912892928} a^{15} - \frac{77799424602646228316903019021077}{148742268715640063595016247420912892928} a^{14} + \frac{59710247126172652450020253368029397}{37185567178910015898754061855228223232} a^{13} - \frac{539457597260070044705291547423287717}{148742268715640063595016247420912892928} a^{12} + \frac{8680715502099908704956755006265061945}{148742268715640063595016247420912892928} a^{11} + \frac{15535072470016155848189280244442424543}{148742268715640063595016247420912892928} a^{10} - \frac{13556613322108277694419905025902296299}{74371134357820031797508123710456446464} a^{9} - \frac{15149588044067563809529033199759899991}{148742268715640063595016247420912892928} a^{8} + \frac{63145431660925432771694040370734918615}{148742268715640063595016247420912892928} a^{7} + \frac{7628941695899077981670901636207123985}{74371134357820031797508123710456446464} a^{6} + \frac{1631182660791769906289665804716788937}{18592783589455007949377030927614111616} a^{5} + \frac{912609898124737580987524821326361989}{2324097948681875993672128865951763952} a^{4} + \frac{1226840856983463319786094837319643751}{9296391794727503974688515463807055808} a^{3} + \frac{99005469206000971641253138339346493}{581024487170468998418032216487940988} a^{2} - \frac{218033789841340978904497722092107179}{581024487170468998418032216487940988} a + \frac{473128043561555809212484712268360}{3925841129530195935256974435729331}$

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

Class group and class number

$C_{33}$, which has order $33$ (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:  \( 4516212.78969 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.0.5031233.1, 4.0.122713.1, 8.0.194754273881.1, 8.8.1037845525511849.1, 8.0.25313305500289.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
41Data not computed
$73$73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.8.4.1$x^{8} + 138554 x^{4} - 389017 x^{2} + 4799302729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$