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Properties

Label 16.0.95171731360...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 29^{6}$
Root discriminant $23.64$
Ramified primes $2, 5, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, -140, 490, -950, 1151, -930, 697, -494, 204, -86, 34, 12, 22, -2, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 9*x^14 - 2*x^13 + 22*x^12 + 12*x^11 + 34*x^10 - 86*x^9 + 204*x^8 - 494*x^7 + 697*x^6 - 930*x^5 + 1151*x^4 - 950*x^3 + 490*x^2 - 140*x + 20)
 
gp: K = bnfinit(x^16 - 2*x^15 + 9*x^14 - 2*x^13 + 22*x^12 + 12*x^11 + 34*x^10 - 86*x^9 + 204*x^8 - 494*x^7 + 697*x^6 - 930*x^5 + 1151*x^4 - 950*x^3 + 490*x^2 - 140*x + 20, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 9 x^{14} - 2 x^{13} + 22 x^{12} + 12 x^{11} + 34 x^{10} - 86 x^{9} + 204 x^{8} - 494 x^{7} + 697 x^{6} - 930 x^{5} + 1151 x^{4} - 950 x^{3} + 490 x^{2} - 140 x + 20 \)

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:  \(9517173136000000000000=2^{16}\cdot 5^{12}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.64$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{1410} a^{14} + \frac{5}{282} a^{13} - \frac{41}{1410} a^{12} + \frac{112}{705} a^{11} + \frac{61}{705} a^{10} - \frac{23}{705} a^{9} - \frac{91}{235} a^{8} - \frac{3}{47} a^{7} - \frac{102}{235} a^{6} - \frac{136}{705} a^{5} + \frac{11}{30} a^{4} - \frac{85}{282} a^{3} + \frac{23}{94} a^{2} + \frac{65}{141} a - \frac{14}{141}$, $\frac{1}{55599024033990} a^{15} + \frac{12566562971}{55599024033990} a^{14} - \frac{332430395846}{27799512016995} a^{13} + \frac{12392634632}{3088834668555} a^{12} + \frac{1257484948391}{27799512016995} a^{11} - \frac{8434864395331}{27799512016995} a^{10} + \frac{9412602947401}{27799512016995} a^{9} + \frac{844483899088}{9266504005665} a^{8} + \frac{2789300192261}{9266504005665} a^{7} - \frac{6050312907784}{27799512016995} a^{6} + \frac{19113485088407}{55599024033990} a^{5} - \frac{3891647739829}{55599024033990} a^{4} + \frac{1949734676482}{5559902403399} a^{3} - \frac{645675739660}{5559902403399} a^{2} + \frac{782865200915}{1853300801133} a + \frac{46172215234}{5559902403399}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{4425489055}{118295795817} a^{15} + \frac{24871156145}{236591591634} a^{14} - \frac{87682687585}{236591591634} a^{13} + \frac{1313303865}{4381325771} a^{12} - \frac{78350775835}{118295795817} a^{11} + \frac{26323753253}{118295795817} a^{10} - \frac{46567222730}{118295795817} a^{9} + \frac{182804499745}{39431931939} a^{8} - \frac{373378581325}{39431931939} a^{7} + \frac{2627943628865}{118295795817} a^{6} - \frac{4344124298117}{118295795817} a^{5} + \frac{10373978337515}{236591591634} a^{4} - \frac{13485302146175}{236591591634} a^{3} + \frac{6127131315880}{118295795817} a^{2} - \frac{943474670930}{39431931939} a + \frac{594669329981}{118295795817} \) (order $4$)
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:  \( 71253.3166668 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.4.58000.1, 4.0.3625.1, \(\Q(i, \sqrt{5})\), 8.0.3364000000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$