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Properties

Label 16.0.20900613426...9312.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{57}\cdot 449^{4}\cdot 1889^{2}$
Root discriminant $139.64$
Ramified primes $2, 449, 1889$
Class number $1360672$ (GRH)
Class group $[2, 2, 2, 2, 85042]$ (GRH)
Galois group 16T1584

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13719758722, -15119234336, 21102481608, -9998030224, 5267824628, -1361814944, 504053312, -82717776, 27814397, -2489256, 970632, -35512, 20314, -192, 224, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 224*x^14 - 192*x^13 + 20314*x^12 - 35512*x^11 + 970632*x^10 - 2489256*x^9 + 27814397*x^8 - 82717776*x^7 + 504053312*x^6 - 1361814944*x^5 + 5267824628*x^4 - 9998030224*x^3 + 21102481608*x^2 - 15119234336*x + 13719758722)
 
gp: K = bnfinit(x^16 + 224*x^14 - 192*x^13 + 20314*x^12 - 35512*x^11 + 970632*x^10 - 2489256*x^9 + 27814397*x^8 - 82717776*x^7 + 504053312*x^6 - 1361814944*x^5 + 5267824628*x^4 - 9998030224*x^3 + 21102481608*x^2 - 15119234336*x + 13719758722, 1)
 

Normalized defining polynomial

\( x^{16} + 224 x^{14} - 192 x^{13} + 20314 x^{12} - 35512 x^{11} + 970632 x^{10} - 2489256 x^{9} + 27814397 x^{8} - 82717776 x^{7} + 504053312 x^{6} - 1361814944 x^{5} + 5267824628 x^{4} - 9998030224 x^{3} + 21102481608 x^{2} - 15119234336 x + 13719758722 \)

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:  \(20900613426398125371268864579469312=2^{57}\cdot 449^{4}\cdot 1889^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.64$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{13} + \frac{3}{7} a^{12} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{15} - \frac{1675689101556181205214396903642832786354663265450430799646506757820730}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{14} + \frac{18631577690028334413218091983511900884691293981265714690838732177815927}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{13} - \frac{75996294990462539431457501662232866603196276541000807471161204616732179}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{12} - \frac{71605459853155892728375277105855828917368457604126132790201297190182701}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{11} - \frac{82125950513514977640098857374643763511657936822239505304947407090176580}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{10} + \frac{1222802031115757658671115958445318433077751990641667116890356351963840}{28224049510532400495959459350492709458324126939769003330846985929527433} a^{9} + \frac{9546117615879351817299222271669541131580641936421159563966517170590691}{28224049510532400495959459350492709458324126939769003330846985929527433} a^{8} + \frac{29089119917633018288899959606477074089012792108262446090883623182869130}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{7} - \frac{5815146172159997825457566776033053266348103666246154070179711114602546}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{6} - \frac{74322287751798943721643968199061407720433994521558372941760368941879700}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{5} - \frac{35165014984468294567995259829365365521633049602855890198677886119340553}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{4} + \frac{67915032147048205173359232575989947871035749364283906275352877069823341}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{3} + \frac{73613655870837287382672308433133655190870659127606758782179052837222105}{197568346573726803471716215453448966208268888578383023315928901506692031} a^{2} + \frac{13950869963263990012101627703848583725512654250776191671747771510877945}{197568346573726803471716215453448966208268888578383023315928901506692031} a + \frac{41091323376903818806073721047977312307689443926218314228087299542485808}{197568346573726803471716215453448966208268888578383023315928901506692031}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{85042}$, which has order $1360672$ (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:  \( 56271.9156358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1584:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1584 are not computed
Character table for t16n1584 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.31.17$x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8:C_2$$[2, 3, 4, 5]$
2.8.26.4$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
449Data not computed
1889Data not computed