↑

Properties

Label 16.4.76087965578...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{40}\cdot 5^{8}\cdot 11^{6}$
Root discriminant $31.09$
Ramified primes $2, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-361, 1748, -1760, -636, 3652, -8468, 8420, -3428, -257, 428, 184, -356, 266, -120, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 120*x^13 + 266*x^12 - 356*x^11 + 184*x^10 + 428*x^9 - 257*x^8 - 3428*x^7 + 8420*x^6 - 8468*x^5 + 3652*x^4 - 636*x^3 - 1760*x^2 + 1748*x - 361)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 120*x^13 + 266*x^12 - 356*x^11 + 184*x^10 + 428*x^9 - 257*x^8 - 3428*x^7 + 8420*x^6 - 8468*x^5 + 3652*x^4 - 636*x^3 - 1760*x^2 + 1748*x - 361, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 120 x^{13} + 266 x^{12} - 356 x^{11} + 184 x^{10} + 428 x^{9} - 257 x^{8} - 3428 x^{7} + 8420 x^{6} - 8468 x^{5} + 3652 x^{4} - 636 x^{3} - 1760 x^{2} + 1748 x - 361 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(760879655786905600000000=2^{40}\cdot 5^{8}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.09$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{5}{11} a^{11} + \frac{3}{11} a^{10} + \frac{5}{11} a^{9} + \frac{1}{11} a^{8} - \frac{5}{11} a^{7} + \frac{2}{11} a^{6} + \frac{4}{11} a^{5} - \frac{2}{11} a^{4} - \frac{3}{11} a^{3} + \frac{3}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} + \frac{1}{11} a^{8} + \frac{5}{11} a^{7} + \frac{5}{11} a^{6} - \frac{4}{11} a^{4} + \frac{4}{11} a^{3} + \frac{3}{11} a^{2} - \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{55} a^{14} - \frac{1}{55} a^{12} + \frac{7}{55} a^{11} - \frac{27}{55} a^{10} - \frac{4}{55} a^{9} + \frac{4}{55} a^{8} - \frac{1}{55} a^{7} - \frac{24}{55} a^{6} - \frac{19}{55} a^{5} + \frac{17}{55} a^{4} + \frac{17}{55} a^{3} - \frac{24}{55} a^{2} + \frac{4}{11} a + \frac{9}{55}$, $\frac{1}{32116213727653675355468755} a^{15} - \frac{162673426880001149120686}{32116213727653675355468755} a^{14} - \frac{961091226816320849505831}{32116213727653675355468755} a^{13} + \frac{263743822696574500141088}{32116213727653675355468755} a^{12} - \frac{17751483198503480954529}{58287139251640064166005} a^{11} + \frac{6968671420356251765500943}{32116213727653675355468755} a^{10} - \frac{4414531956708513702843197}{32116213727653675355468755} a^{9} - \frac{2403453187194669788866821}{6423242745530735071093751} a^{8} + \frac{325819718108211677736813}{1107455645781161219154095} a^{7} + \frac{1498487603418088210621257}{6423242745530735071093751} a^{6} + \frac{7370251306773915214425046}{32116213727653675355468755} a^{5} - \frac{41341916972032032923880}{6423242745530735071093751} a^{4} + \frac{12695671961084315068978664}{32116213727653675355468755} a^{3} - \frac{1998758492745285256589301}{32116213727653675355468755} a^{2} - \frac{1953403151110469605675056}{32116213727653675355468755} a + \frac{806603725902788506254064}{1690327038297561860814145}$

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:  $9$
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:  \( 525091.187748 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

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 $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1600.1, 4.4.4400.1, 4.2.17600.2, 8.4.4956160000.6

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$