↑

Properties

Label 16.8.11420589882...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{38}\cdot 5^{8}\cdot 89^{4}\cdot 130201^{2}$
Root discriminant $155.28$
Ramified primes $2, 5, 89, 130201$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1605

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67809201604, 0, -13483094756, 0, -1120383490, 0, 161587388, 0, 7092685, 0, -429814, 0, -13001, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 - 13001*x^12 - 429814*x^10 + 7092685*x^8 + 161587388*x^6 - 1120383490*x^4 - 13483094756*x^2 + 67809201604)
 
gp: K = bnfinit(x^16 - 18*x^14 - 13001*x^12 - 429814*x^10 + 7092685*x^8 + 161587388*x^6 - 1120383490*x^4 - 13483094756*x^2 + 67809201604, 1)
 

Normalized defining polynomial

\( x^{16} - 18 x^{14} - 13001 x^{12} - 429814 x^{10} + 7092685 x^{8} + 161587388 x^{6} - 1120383490 x^{4} - 13483094756 x^{2} + 67809201604 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(114205898821156016581142118400000000=2^{38}\cdot 5^{8}\cdot 89^{4}\cdot 130201^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.28$
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:  $2, 5, 89, 130201$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{388058918244009834285924187419742689458} a^{14} - \frac{57551510097577453348629919856715942565}{388058918244009834285924187419742689458} a^{12} - \frac{172991091723919640068085038465777453081}{388058918244009834285924187419742689458} a^{10} - \frac{83803835986456205983669567539420010565}{388058918244009834285924187419742689458} a^{8} + \frac{127604181784849044883217702054867850913}{388058918244009834285924187419742689458} a^{6} + \frac{48483605726627002257210319254095341027}{388058918244009834285924187419742689458} a^{4} - \frac{56197784368620715602561574380593764864}{194029459122004917142962093709871344729} a^{2} + \frac{629742371059509165082936854553455}{1490230175820499974216496752788929}$, $\frac{1}{388058918244009834285924187419742689458} a^{15} - \frac{57551510097577453348629919856715942565}{388058918244009834285924187419742689458} a^{13} - \frac{172991091723919640068085038465777453081}{388058918244009834285924187419742689458} a^{11} - \frac{83803835986456205983669567539420010565}{388058918244009834285924187419742689458} a^{9} + \frac{127604181784849044883217702054867850913}{388058918244009834285924187419742689458} a^{7} + \frac{48483605726627002257210319254095341027}{388058918244009834285924187419742689458} a^{5} - \frac{56197784368620715602561574380593764864}{194029459122004917142962093709871344729} a^{3} + \frac{629742371059509165082936854553455}{1490230175820499974216496752788929} a$

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:  $11$
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:  \( 525423903410 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.5069440000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.18.48$x^{8} + 6 x^{6} + 2 x^{4} + 12$$4$$2$$18$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
2.8.20.42$x^{8} + 16 x^{7} + 8 x^{6} + 240$$4$$2$$20$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
5Data not computed
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
130201Data not computed