↑

Properties

Label 16.0.69052763263...1909.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{10}\cdot 61^{9}$
Root discriminant $20.07$
Ramified primes $3, 61$
Class number $2$
Class group $[2]$
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 357, 1129, 1749, 885, -933, -1152, 251, 715, 28, -213, 20, 72, 5, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 9*x^14 + 5*x^13 + 72*x^12 + 20*x^11 - 213*x^10 + 28*x^9 + 715*x^8 + 251*x^7 - 1152*x^6 - 933*x^5 + 885*x^4 + 1749*x^3 + 1129*x^2 + 357*x + 49)
 
gp: K = bnfinit(x^16 - x^15 - 9*x^14 + 5*x^13 + 72*x^12 + 20*x^11 - 213*x^10 + 28*x^9 + 715*x^8 + 251*x^7 - 1152*x^6 - 933*x^5 + 885*x^4 + 1749*x^3 + 1129*x^2 + 357*x + 49, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 9 x^{14} + 5 x^{13} + 72 x^{12} + 20 x^{11} - 213 x^{10} + 28 x^{9} + 715 x^{8} + 251 x^{7} - 1152 x^{6} - 933 x^{5} + 885 x^{4} + 1749 x^{3} + 1129 x^{2} + 357 x + 49 \)

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:  \(690527632635763191909=3^{10}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.07$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{185302770705367} a^{15} + \frac{50975529045342}{185302770705367} a^{14} + \frac{74819628991581}{185302770705367} a^{13} + \frac{63341673171684}{185302770705367} a^{12} - \frac{35380027107290}{185302770705367} a^{11} + \frac{52534591978392}{185302770705367} a^{10} + \frac{90061362421517}{185302770705367} a^{9} - \frac{8133267128612}{26471824386481} a^{8} - \frac{75741890586557}{185302770705367} a^{7} + \frac{73220213351333}{185302770705367} a^{6} + \frac{45679626691464}{185302770705367} a^{5} - \frac{64432872484185}{185302770705367} a^{4} - \frac{23005277833843}{185302770705367} a^{3} - \frac{86691235534219}{185302770705367} a^{2} + \frac{63919366260073}{185302770705367} a - \frac{10436393348520}{26471824386481}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{2069344252469}{335086384639} a^{15} - \frac{3656190098175}{335086384639} a^{14} - \frac{15814650577331}{335086384639} a^{13} + \frac{22460250914995}{335086384639} a^{12} + \frac{131734802354286}{335086384639} a^{11} - \frac{59550285570297}{335086384639} a^{10} - \frac{394793263072470}{335086384639} a^{9} + \frac{360336301537665}{335086384639} a^{8} + \frac{1202419164021716}{335086384639} a^{7} - \frac{401179401710838}{335086384639} a^{6} - \frac{2073876302424473}{335086384639} a^{5} - \frac{342714145898392}{335086384639} a^{4} + \frac{2090104439530065}{335086384639} a^{3} + \frac{2017737332666502}{335086384639} a^{2} + \frac{793804285440229}{335086384639} a + \frac{133263530380705}{335086384639} \) (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:  \( 7625.28778109 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.549.1, 8.0.18385461.1

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 $16$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16$ $16$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.8.4.1$x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$