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Properties

Label 16.4.51362029671...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{28}\cdot 5^{10}\cdot 241^{6}$
Root discriminant $71.93$
Ramified primes $2, 5, 241$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-128476, 62824, 95148, 145832, 82950, 73664, 10910, 2512, -11879, -5484, -4824, -1304, -635, -92, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 - 92*x^13 - 635*x^12 - 1304*x^11 - 4824*x^10 - 5484*x^9 - 11879*x^8 + 2512*x^7 + 10910*x^6 + 73664*x^5 + 82950*x^4 + 145832*x^3 + 95148*x^2 + 62824*x - 128476)
 
gp: K = bnfinit(x^16 - 18*x^14 - 92*x^13 - 635*x^12 - 1304*x^11 - 4824*x^10 - 5484*x^9 - 11879*x^8 + 2512*x^7 + 10910*x^6 + 73664*x^5 + 82950*x^4 + 145832*x^3 + 95148*x^2 + 62824*x - 128476, 1)
 

Normalized defining polynomial

\( x^{16} - 18 x^{14} - 92 x^{13} - 635 x^{12} - 1304 x^{11} - 4824 x^{10} - 5484 x^{9} - 11879 x^{8} + 2512 x^{7} + 10910 x^{6} + 73664 x^{5} + 82950 x^{4} + 145832 x^{3} + 95148 x^{2} + 62824 x - 128476 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(513620296716624855040000000000=2^{28}\cdot 5^{10}\cdot 241^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.93$
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:  $2, 5, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{10} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{1560} a^{14} + \frac{43}{1560} a^{13} - \frac{1}{13} a^{12} + \frac{59}{312} a^{11} + \frac{7}{52} a^{10} + \frac{7}{520} a^{9} + \frac{289}{1560} a^{8} + \frac{73}{390} a^{7} - \frac{1}{780} a^{6} - \frac{233}{780} a^{5} - \frac{1}{60} a^{4} - \frac{71}{195} a^{3} - \frac{3}{10} a^{2} + \frac{109}{390} a - \frac{3}{130}$, $\frac{1}{202540862721557896315869970247682120} a^{15} - \frac{9757148418256241124969153334009}{50635215680389474078967492561920530} a^{14} - \frac{3111717893441014605075143898199179}{67513620907185965438623323415894040} a^{13} + \frac{233188254422146247529524359832877}{1038671090879784083671128052552216} a^{12} - \frac{3324546201979567456904850923213611}{40508172544311579263173994049536424} a^{11} - \frac{27298379003406464951866836294730909}{202540862721557896315869970247682120} a^{10} + \frac{3407755739132061332150881160941543}{20254086272155789631586997024768212} a^{9} + \frac{479196624354797631791574739641701}{3320342011828817972391310987666920} a^{8} + \frac{4681990784243190926970664970788123}{20254086272155789631586997024768212} a^{7} - \frac{5712374385736782516724861113448361}{25317607840194737039483746280960265} a^{6} + \frac{10782246650009413082909644564311917}{50635215680389474078967492561920530} a^{5} + \frac{1454212328483020852819960912938083}{101270431360778948157934985123841060} a^{4} + \frac{2701963238064389083274546347549651}{50635215680389474078967492561920530} a^{3} - \frac{4605524035311844598365234610476589}{25317607840194737039483746280960265} a^{2} + \frac{1979571769396147949903573933611525}{5063521568038947407896749256192053} a - \frac{19445186713158565534274692259937739}{50635215680389474078967492561920530}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 251243993.879 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), 4.4.385600.1, 4.4.385600.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.148687360000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
241Data not computed