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Properties

Label 16.6.10009146154...6143.4
Degree $16$
Signature $[6, 5]$
Discriminant $-\,23^{5}\cdot 41^{15}$
Root discriminant $86.60$
Ramified primes $23, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-235652, 358976, -365215, -38060, 150097, -168868, 70466, -1146, -15113, 12800, -5046, 1278, -122, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 122*x^12 + 1278*x^11 - 5046*x^10 + 12800*x^9 - 15113*x^8 - 1146*x^7 + 70466*x^6 - 168868*x^5 + 150097*x^4 - 38060*x^3 - 365215*x^2 + 358976*x - 235652)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 122*x^12 + 1278*x^11 - 5046*x^10 + 12800*x^9 - 15113*x^8 - 1146*x^7 + 70466*x^6 - 168868*x^5 + 150097*x^4 - 38060*x^3 - 365215*x^2 + 358976*x - 235652, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 122 x^{12} + 1278 x^{11} - 5046 x^{10} + 12800 x^{9} - 15113 x^{8} - 1146 x^{7} + 70466 x^{6} - 168868 x^{5} + 150097 x^{4} - 38060 x^{3} - 365215 x^{2} + 358976 x - 235652 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10009146154007580091982470826143=-\,23^{5}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.60$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{92} a^{12} - \frac{3}{46} a^{11} - \frac{9}{46} a^{10} + \frac{7}{92} a^{9} + \frac{9}{46} a^{8} + \frac{1}{23} a^{7} - \frac{9}{92} a^{6} + \frac{11}{23} a^{5} + \frac{17}{92} a^{4} + \frac{19}{92} a^{3} + \frac{21}{46} a^{2} - \frac{27}{92} a - \frac{21}{46}$, $\frac{1}{92} a^{13} - \frac{2}{23} a^{11} - \frac{9}{92} a^{10} + \frac{7}{46} a^{9} + \frac{5}{23} a^{8} - \frac{31}{92} a^{7} - \frac{5}{46} a^{6} + \frac{5}{92} a^{5} - \frac{17}{92} a^{4} + \frac{9}{46} a^{3} - \frac{5}{92} a^{2} - \frac{5}{23} a + \frac{6}{23}$, $\frac{1}{2373131349532928} a^{14} - \frac{7}{2373131349532928} a^{13} + \frac{792743109133}{593282837383232} a^{12} - \frac{19025834619101}{2373131349532928} a^{11} - \frac{501780481245555}{2373131349532928} a^{10} + \frac{155087270351553}{1186565674766464} a^{9} + \frac{401405616906053}{2373131349532928} a^{8} + \frac{1070308806413183}{2373131349532928} a^{7} + \frac{481923705640391}{2373131349532928} a^{6} - \frac{52004816760309}{148320709345808} a^{5} - \frac{810821270080003}{2373131349532928} a^{4} + \frac{1159008739906249}{2373131349532928} a^{3} + \frac{55866785574595}{2373131349532928} a^{2} + \frac{263744209015607}{593282837383232} a + \frac{98821363384405}{593282837383232}$, $\frac{1}{329865257585076992} a^{15} + \frac{31}{164932628792538496} a^{14} - \frac{357957711188527}{329865257585076992} a^{13} - \frac{1038384223211225}{329865257585076992} a^{12} + \frac{642642736369439}{82466314396269248} a^{11} + \frac{18515288767840643}{329865257585076992} a^{10} + \frac{48423791889746255}{329865257585076992} a^{9} - \frac{8250398521568397}{41233157198134624} a^{8} + \frac{10030374590285441}{164932628792538496} a^{7} - \frac{143861728799717421}{329865257585076992} a^{6} - \frac{148919028375176083}{329865257585076992} a^{5} - \frac{4591132567512323}{164932628792538496} a^{4} - \frac{7628113972176105}{20616578599067312} a^{3} + \frac{53197849022509931}{329865257585076992} a^{2} - \frac{3821409697591975}{10308289299533656} a - \frac{883372478707201}{3585491930272576}$

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:  $10$
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:  \( 1985871420.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.4479348299263.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41Data not computed