↑

Properties

Label 16.0.76383686320...8576.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{10}\cdot 7^{6}$
Root discriminant $23.32$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 24, 76, 128, 270, 864, 1852, 1888, 484, -584, -292, 192, 134, -16, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 - 16*x^13 + 134*x^12 + 192*x^11 - 292*x^10 - 584*x^9 + 484*x^8 + 1888*x^7 + 1852*x^6 + 864*x^5 + 270*x^4 + 128*x^3 + 76*x^2 + 24*x + 3)
 
gp: K = bnfinit(x^16 - 20*x^14 - 16*x^13 + 134*x^12 + 192*x^11 - 292*x^10 - 584*x^9 + 484*x^8 + 1888*x^7 + 1852*x^6 + 864*x^5 + 270*x^4 + 128*x^3 + 76*x^2 + 24*x + 3, 1)
 

Normalized defining polynomial

\( x^{16} - 20 x^{14} - 16 x^{13} + 134 x^{12} + 192 x^{11} - 292 x^{10} - 584 x^{9} + 484 x^{8} + 1888 x^{7} + 1852 x^{6} + 864 x^{5} + 270 x^{4} + 128 x^{3} + 76 x^{2} + 24 x + 3 \)

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:  \(7638368632008213528576=2^{40}\cdot 3^{10}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.32$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{2532} a^{14} - \frac{247}{2532} a^{13} - \frac{17}{633} a^{12} + \frac{44}{633} a^{11} - \frac{265}{2532} a^{10} - \frac{409}{2532} a^{9} + \frac{19}{633} a^{8} - \frac{134}{633} a^{7} + \frac{137}{2532} a^{6} - \frac{1115}{2532} a^{5} - \frac{86}{633} a^{4} + \frac{262}{633} a^{3} + \frac{979}{2532} a^{2} - \frac{59}{844} a - \frac{94}{211}$, $\frac{1}{28891899996} a^{15} + \frac{1994413}{14445949998} a^{14} + \frac{1729884929}{14445949998} a^{13} + \frac{991947799}{14445949998} a^{12} - \frac{1333480507}{28891899996} a^{11} - \frac{1412936662}{7222974999} a^{10} - \frac{1606229396}{7222974999} a^{9} - \frac{741589886}{7222974999} a^{8} + \frac{2659946369}{28891899996} a^{7} + \frac{6945802691}{14445949998} a^{6} - \frac{4861588669}{14445949998} a^{5} + \frac{3576612743}{14445949998} a^{4} + \frac{8078111521}{28891899996} a^{3} - \frac{789047799}{2407658333} a^{2} + \frac{70099645}{2407658333} a - \frac{567237328}{2407658333}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{541070}{1476789} a^{15} + \frac{379559}{2953578} a^{14} + \frac{10728551}{1476789} a^{13} + \frac{19677655}{5907156} a^{12} - \frac{24577104}{492263} a^{11} - \frac{52051541}{984526} a^{10} + \frac{181972079}{1476789} a^{9} + \frac{333804165}{1969052} a^{8} - \frac{340683350}{1476789} a^{7} - \frac{1790891273}{2953578} a^{6} - \frac{235378201}{492263} a^{5} - \frac{335371119}{1969052} a^{4} - \frac{25892788}{492263} a^{3} - \frac{97048151}{2953578} a^{2} - \frac{9343463}{492263} a - \frac{6237591}{1969052} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 255980.06537 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), 4.0.1008.1, 4.0.4032.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.0.260112384.6

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$