↑

Properties

Label 16.0.47557574440...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 29^{4}\cdot 41^{2}$
Root discriminant $30.19$
Ramified primes $2, 5, 29, 41$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T610)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1181, 296, -10, -2830, 1320, -2792, 2563, -910, 1425, -240, 373, -82, 75, -20, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 15*x^14 - 20*x^13 + 75*x^12 - 82*x^11 + 373*x^10 - 240*x^9 + 1425*x^8 - 910*x^7 + 2563*x^6 - 2792*x^5 + 1320*x^4 - 2830*x^3 - 10*x^2 + 296*x + 1181)
 
gp: K = bnfinit(x^16 - 4*x^15 + 15*x^14 - 20*x^13 + 75*x^12 - 82*x^11 + 373*x^10 - 240*x^9 + 1425*x^8 - 910*x^7 + 2563*x^6 - 2792*x^5 + 1320*x^4 - 2830*x^3 - 10*x^2 + 296*x + 1181, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 15 x^{14} - 20 x^{13} + 75 x^{12} - 82 x^{11} + 373 x^{10} - 240 x^{9} + 1425 x^{8} - 910 x^{7} + 2563 x^{6} - 2792 x^{5} + 1320 x^{4} - 2830 x^{3} - 10 x^{2} + 296 x + 1181 \)

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:  \(475575744400000000000000=2^{16}\cdot 5^{14}\cdot 29^{4}\cdot 41^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.19$
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:  $2, 5, 29, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{55} a^{13} - \frac{2}{55} a^{12} - \frac{1}{11} a^{11} - \frac{3}{55} a^{10} + \frac{1}{11} a^{9} + \frac{1}{55} a^{8} + \frac{13}{55} a^{7} - \frac{13}{55} a^{6} - \frac{2}{55} a^{5} + \frac{2}{11} a^{4} - \frac{3}{11} a^{3} + \frac{4}{11} a^{2} - \frac{23}{55} a - \frac{19}{55}$, $\frac{1}{605} a^{14} + \frac{1}{605} a^{13} + \frac{4}{55} a^{12} - \frac{18}{605} a^{11} - \frac{48}{605} a^{10} - \frac{28}{605} a^{9} + \frac{49}{605} a^{8} + \frac{59}{605} a^{7} - \frac{50}{121} a^{6} - \frac{139}{605} a^{5} - \frac{139}{605} a^{4} - \frac{47}{605} a^{3} + \frac{58}{121} a^{2} + \frac{3}{55} a - \frac{266}{605}$, $\frac{1}{10340053856701135655} a^{15} + \frac{2767956314606509}{10340053856701135655} a^{14} - \frac{34872307319950691}{10340053856701135655} a^{13} + \frac{180398197856388912}{10340053856701135655} a^{12} + \frac{912775968446610826}{10340053856701135655} a^{11} - \frac{143754944265160587}{10340053856701135655} a^{10} - \frac{424649189768977791}{10340053856701135655} a^{9} - \frac{89110477770723249}{2068010771340227131} a^{8} + \frac{4898349721157369349}{10340053856701135655} a^{7} - \frac{514718591475407653}{2068010771340227131} a^{6} - \frac{2538593600211779193}{10340053856701135655} a^{5} - \frac{45928020893946594}{2068010771340227131} a^{4} + \frac{1901250661190588296}{10340053856701135655} a^{3} - \frac{3386210035601909273}{10340053856701135655} a^{2} - \frac{2680265798511932556}{10340053856701135655} a - \frac{790601433888623818}{2068010771340227131}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 12061.7171318 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T610):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.725.1, 4.4.58000.1, 8.8.3364000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41Data not computed