↑

Properties

Label 16.0.62847496190...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{8}\cdot 229^{3}$
Root discriminant $30.72$
Ramified primes $5, 11, 229$
Class number $4$
Class group $[2, 2]$
Galois group 16T1477

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7951, 14581, 1172, -568, 4652, -4000, 123, 779, -520, -134, 103, 10, 37, 8, -13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 13*x^14 + 8*x^13 + 37*x^12 + 10*x^11 + 103*x^10 - 134*x^9 - 520*x^8 + 779*x^7 + 123*x^6 - 4000*x^5 + 4652*x^4 - 568*x^3 + 1172*x^2 + 14581*x + 7951)
 
gp: K = bnfinit(x^16 - x^15 - 13*x^14 + 8*x^13 + 37*x^12 + 10*x^11 + 103*x^10 - 134*x^9 - 520*x^8 + 779*x^7 + 123*x^6 - 4000*x^5 + 4652*x^4 - 568*x^3 + 1172*x^2 + 14581*x + 7951, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 13 x^{14} + 8 x^{13} + 37 x^{12} + 10 x^{11} + 103 x^{10} - 134 x^{9} - 520 x^{8} + 779 x^{7} + 123 x^{6} - 4000 x^{5} + 4652 x^{4} - 568 x^{3} + 1172 x^{2} + 14581 x + 7951 \)

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:  \(628474961909499267578125=5^{12}\cdot 11^{8}\cdot 229^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.72$
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:  $5, 11, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{12} - \frac{3}{11} a^{11} + \frac{5}{11} a^{10} - \frac{5}{11} a^{9} - \frac{2}{11} a^{8} + \frac{3}{11} a^{7} + \frac{2}{11} a^{6} - \frac{1}{11} a^{5} - \frac{5}{11} a^{4} + \frac{3}{11} a^{3} - \frac{3}{11} a^{2} + \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{12} - \frac{4}{11} a^{11} - \frac{1}{11} a^{10} + \frac{5}{11} a^{9} - \frac{3}{11} a^{8} + \frac{5}{11} a^{6} + \frac{3}{11} a^{5} - \frac{1}{11} a^{4} - \frac{5}{11} a^{3} + \frac{4}{11} a^{2} + \frac{3}{11} a + \frac{2}{11}$, $\frac{1}{8713725002569715766705747119} a^{15} - \frac{247927380680457466919243020}{8713725002569715766705747119} a^{14} - \frac{249926916811072903092689593}{8713725002569715766705747119} a^{13} - \frac{382481001261008225328698349}{792156818415428706064158829} a^{12} + \frac{897381300508392690796666832}{8713725002569715766705747119} a^{11} + \frac{1836782353635229789022202505}{8713725002569715766705747119} a^{10} - \frac{3545211909524204227656831748}{8713725002569715766705747119} a^{9} - \frac{498702777287281749940119603}{8713725002569715766705747119} a^{8} + \frac{2222508036078815190343243281}{8713725002569715766705747119} a^{7} - \frac{1062596602523549767854739675}{8713725002569715766705747119} a^{6} - \frac{2267655937697078311676340401}{8713725002569715766705747119} a^{5} + \frac{2926360499015515189486333353}{8713725002569715766705747119} a^{4} - \frac{1668059329553950408463622606}{8713725002569715766705747119} a^{3} - \frac{811386134704449891037609237}{8713725002569715766705747119} a^{2} - \frac{285697516278945685027281156}{8713725002569715766705747119} a + \frac{1548280166234950202581382705}{8713725002569715766705747119}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 32102.9275967 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1477:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1477 are not computed
Character table for t16n1477 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.0.2095493125.3

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ R $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
229Data not computed