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Properties

Label 16.4.42345450969...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{8}\cdot 89^{2}$
Root discriminant $39.96$
Ramified primes $2, 5, 13, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1123

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4684, 22288, 81666, -160, -44765, 122178, -120353, 81372, -41895, 17048, -5559, 1140, -47, -82, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 82*x^13 - 47*x^12 + 1140*x^11 - 5559*x^10 + 17048*x^9 - 41895*x^8 + 81372*x^7 - 120353*x^6 + 122178*x^5 - 44765*x^4 - 160*x^3 + 81666*x^2 + 22288*x - 4684)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 82*x^13 - 47*x^12 + 1140*x^11 - 5559*x^10 + 17048*x^9 - 41895*x^8 + 81372*x^7 - 120353*x^6 + 122178*x^5 - 44765*x^4 - 160*x^3 + 81666*x^2 + 22288*x - 4684, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 82 x^{13} - 47 x^{12} + 1140 x^{11} - 5559 x^{10} + 17048 x^{9} - 41895 x^{8} + 81372 x^{7} - 120353 x^{6} + 122178 x^{5} - 44765 x^{4} - 160 x^{3} + 81666 x^{2} + 22288 x - 4684 \)

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:  \(42345450969766297600000000=2^{24}\cdot 5^{8}\cdot 13^{8}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.96$
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:  $2, 5, 13, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{22} a^{13} - \frac{3}{22} a^{12} - \frac{1}{11} a^{11} + \frac{1}{11} a^{10} + \frac{5}{11} a^{8} - \frac{3}{22} a^{7} - \frac{5}{22} a^{6} + \frac{1}{22} a^{5} - \frac{5}{22} a^{4} - \frac{5}{11} a^{3} - \frac{4}{11} a^{2} + \frac{2}{11} a - \frac{4}{11}$, $\frac{1}{484} a^{14} + \frac{5}{242} a^{13} - \frac{107}{484} a^{12} + \frac{43}{242} a^{11} + \frac{1}{121} a^{10} + \frac{27}{242} a^{9} + \frac{171}{484} a^{8} - \frac{5}{11} a^{7} + \frac{17}{121} a^{6} + \frac{57}{121} a^{5} - \frac{31}{484} a^{4} + \frac{15}{121} a^{3} + \frac{52}{121} a^{2} + \frac{4}{11} a - \frac{4}{121}$, $\frac{1}{422555832295220613479308279605569128} a^{15} + \frac{362290994855746329803166671566721}{422555832295220613479308279605569128} a^{14} + \frac{2861108286069146575482424398045965}{422555832295220613479308279605569128} a^{13} + \frac{28757269505115114118949779374952831}{422555832295220613479308279605569128} a^{12} - \frac{31844525048909974149982361281658}{208772644414634690454203695457297} a^{11} + \frac{694872960028252147930868832772168}{52819479036902576684913534950696141} a^{10} + \frac{29066403705979578669619204909543625}{422555832295220613479308279605569128} a^{9} - \frac{98284716267655313623188832889947191}{422555832295220613479308279605569128} a^{8} + \frac{81139335243666209820542884391157969}{211277916147610306739654139802784564} a^{7} + \frac{19963975451859282918577641003072697}{211277916147610306739654139802784564} a^{6} - \frac{27339194996995854513649852407374231}{422555832295220613479308279605569128} a^{5} + \frac{87820962431306920056678666713778083}{422555832295220613479308279605569128} a^{4} + \frac{61214891529686796244272438129320787}{211277916147610306739654139802784564} a^{3} + \frac{38202555716444343883723547459016017}{211277916147610306739654139802784564} a^{2} + \frac{44092581244287507272766354698468641}{105638958073805153369827069901392282} a - \frac{12050956192772715281900004917627359}{105638958073805153369827069901392282}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1760126.87636 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1123:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1123 are not computed
Character table for t16n1123 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 8.8.1142440000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$