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Properties

Label 16.4.10704914143...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{32}\cdot 5^{10}\cdot 761^{5}$
Root discriminant $86.97$
Ramified primes $2, 5, 761$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1360

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![72311951, -166669624, 125361914, -70355216, -37254054, 33988936, -25810424, 6678928, -2155256, 177724, -56396, -1976, 1609, -56, 90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 90*x^14 - 56*x^13 + 1609*x^12 - 1976*x^11 - 56396*x^10 + 177724*x^9 - 2155256*x^8 + 6678928*x^7 - 25810424*x^6 + 33988936*x^5 - 37254054*x^4 - 70355216*x^3 + 125361914*x^2 - 166669624*x + 72311951)
 
gp: K = bnfinit(x^16 + 90*x^14 - 56*x^13 + 1609*x^12 - 1976*x^11 - 56396*x^10 + 177724*x^9 - 2155256*x^8 + 6678928*x^7 - 25810424*x^6 + 33988936*x^5 - 37254054*x^4 - 70355216*x^3 + 125361914*x^2 - 166669624*x + 72311951, 1)
 

Normalized defining polynomial

\( x^{16} + 90 x^{14} - 56 x^{13} + 1609 x^{12} - 1976 x^{11} - 56396 x^{10} + 177724 x^{9} - 2155256 x^{8} + 6678928 x^{7} - 25810424 x^{6} + 33988936 x^{5} - 37254054 x^{4} - 70355216 x^{3} + 125361914 x^{2} - 166669624 x + 72311951 \)

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:  \(10704914143082750935040000000000=2^{32}\cdot 5^{10}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.97$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4645} a^{14} - \frac{1866}{4645} a^{13} - \frac{1303}{4645} a^{12} + \frac{1586}{4645} a^{11} - \frac{253}{929} a^{10} - \frac{305}{929} a^{9} + \frac{1294}{4645} a^{8} - \frac{204}{929} a^{7} + \frac{2193}{4645} a^{6} - \frac{147}{929} a^{5} - \frac{81}{4645} a^{4} + \frac{1877}{4645} a^{3} - \frac{1272}{4645} a^{2} - \frac{1902}{4645} a - \frac{761}{4645}$, $\frac{1}{729215778093725115779241389745483197291506914788201531494791565} a^{15} + \frac{38815560709669775155124822906359689397596867288814309375846}{729215778093725115779241389745483197291506914788201531494791565} a^{14} - \frac{34080180666827850958437245073228456561913263852323294163299291}{145843155618745023155848277949096639458301382957640306298958313} a^{13} - \frac{66445387445346682741475359103744688065579761055721739126097285}{145843155618745023155848277949096639458301382957640306298958313} a^{12} - \frac{198232131612582380773492046422097238741492277593941490608267513}{729215778093725115779241389745483197291506914788201531494791565} a^{11} + \frac{64766896633815211349192159374283423550185984458301443279259063}{145843155618745023155848277949096639458301382957640306298958313} a^{10} - \frac{273430284204121689798173978432312423297185994411005670322871856}{729215778093725115779241389745483197291506914788201531494791565} a^{9} + \frac{332595693128221575012932874965808430428277499980269608364776373}{729215778093725115779241389745483197291506914788201531494791565} a^{8} - \frac{279305317909500345860728395205773548458339707534698396902480642}{729215778093725115779241389745483197291506914788201531494791565} a^{7} - \frac{187244623751528606902821337508553996173370475422125458472336269}{729215778093725115779241389745483197291506914788201531494791565} a^{6} + \frac{168879708842567957893346122561308412327001112807857914365124434}{729215778093725115779241389745483197291506914788201531494791565} a^{5} + \frac{72421211659262039231565906641412955633207461009360407362100378}{145843155618745023155848277949096639458301382957640306298958313} a^{4} + \frac{13148691700488249436779221362569177102412553313504438120905142}{729215778093725115779241389745483197291506914788201531494791565} a^{3} - \frac{144807541383572116309633301650063112565737304923955212207501831}{729215778093725115779241389745483197291506914788201531494791565} a^{2} + \frac{28349836900269621767405147272745993597708747513022091814558744}{145843155618745023155848277949096639458301382957640306298958313} a + \frac{137422306250863182347471893845520047443659503224529550270292893}{729215778093725115779241389745483197291506914788201531494791565}$

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:  \( 292670864.561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1360:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 65 conjugacy class representatives for t16n1360 are not computed
Character table for t16n1360 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$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}$
761Data not computed