↑

Properties

Label 16.0.71241529264...4064.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{68}\cdot 17^{6}$
Root discriminant $55.05$
Ramified primes $2, 17$
Class number $96$ (GRH)
Class group $[2, 2, 24]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![60911, 120768, 100336, -4272, -57820, -31424, 10952, 15392, 2486, -3088, -1152, 368, 244, -16, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 - 16*x^13 + 244*x^12 + 368*x^11 - 1152*x^10 - 3088*x^9 + 2486*x^8 + 15392*x^7 + 10952*x^6 - 31424*x^5 - 57820*x^4 - 4272*x^3 + 100336*x^2 + 120768*x + 60911)
 
gp: K = bnfinit(x^16 - 24*x^14 - 16*x^13 + 244*x^12 + 368*x^11 - 1152*x^10 - 3088*x^9 + 2486*x^8 + 15392*x^7 + 10952*x^6 - 31424*x^5 - 57820*x^4 - 4272*x^3 + 100336*x^2 + 120768*x + 60911, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} - 16 x^{13} + 244 x^{12} + 368 x^{11} - 1152 x^{10} - 3088 x^{9} + 2486 x^{8} + 15392 x^{7} + 10952 x^{6} - 31424 x^{5} - 57820 x^{4} - 4272 x^{3} + 100336 x^{2} + 120768 x + 60911 \)

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:  \(7124152926472086209444184064=2^{68}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.05$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{225743051149320649569620461650241} a^{15} + \frac{6141472473300307809614818727614}{225743051149320649569620461650241} a^{14} - \frac{29378479362834739233898354686553}{225743051149320649569620461650241} a^{13} + \frac{49057025023005079323896557076960}{225743051149320649569620461650241} a^{12} - \frac{40115179019795977322681278093897}{225743051149320649569620461650241} a^{11} + \frac{35570565938779635548792050938690}{225743051149320649569620461650241} a^{10} + \frac{56147917546229481741857852335936}{225743051149320649569620461650241} a^{9} + \frac{64701259651202990289992309594283}{225743051149320649569620461650241} a^{8} - \frac{54547333756030159047579592350589}{225743051149320649569620461650241} a^{7} + \frac{63644006451136622324229933458905}{225743051149320649569620461650241} a^{6} + \frac{49014608232968710921015973493819}{225743051149320649569620461650241} a^{5} - \frac{24053016627687842047263459833662}{225743051149320649569620461650241} a^{4} + \frac{69319166879947420697903008195427}{225743051149320649569620461650241} a^{3} - \frac{61115993178174041170513400704646}{225743051149320649569620461650241} a^{2} - \frac{5722390133860903764314944837238}{13279003008783567621742380097073} a - \frac{2504917534237946495493287356030}{13279003008783567621742380097073}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{24}$, which has order $96$ (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:  \( 384472.338777 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.620622774272.1, 8.0.36507222016.5, 8.0.1318823395328.9

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$