↑

Properties

Label 16.0.27115868648...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 101^{8}\cdot 156506561$
Root discriminant $218.56$
Ramified primes $2, 5, 101, 156506561$
Class number $43091200$ (GRH)
Class group $[2, 2, 4, 2693200]$ (GRH)
Galois group 16T1643

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1596523428761, 0, 1425939962652, 0, 228116949332, 0, 14941870635, 0, 490769716, 0, 8834825, 0, 88673, 0, 466, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 466*x^14 + 88673*x^12 + 8834825*x^10 + 490769716*x^8 + 14941870635*x^6 + 228116949332*x^4 + 1425939962652*x^2 + 1596523428761)
 
gp: K = bnfinit(x^16 + 466*x^14 + 88673*x^12 + 8834825*x^10 + 490769716*x^8 + 14941870635*x^6 + 228116949332*x^4 + 1425939962652*x^2 + 1596523428761, 1)
 

Normalized defining polynomial

\( x^{16} + 466 x^{14} + 88673 x^{12} + 8834825 x^{10} + 490769716 x^{8} + 14941870635 x^{6} + 228116949332 x^{4} + 1425939962652 x^{2} + 1596523428761 \)

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:  \(27115868648582425837365776000000000000=2^{16}\cdot 5^{12}\cdot 101^{8}\cdot 156506561\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $218.56$
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:  $2, 5, 101, 156506561$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1306587198137082201119588050014137574451} a^{14} - \frac{606161139316878335000452691059116161647}{1306587198137082201119588050014137574451} a^{12} - \frac{207719788076672791756570058890084814285}{1306587198137082201119588050014137574451} a^{10} + \frac{502414363853776085947223890243663936134}{1306587198137082201119588050014137574451} a^{8} + \frac{257617575451993487819026914554581468468}{1306587198137082201119588050014137574451} a^{6} + \frac{447529200623941735894547625894569861321}{1306587198137082201119588050014137574451} a^{4} - \frac{492116662567559112172956339576036663195}{1306587198137082201119588050014137574451} a^{2} - \frac{2530830192133113934360196053013280870}{12936506912248338624946416336773639351}$, $\frac{1}{1306587198137082201119588050014137574451} a^{15} - \frac{606161139316878335000452691059116161647}{1306587198137082201119588050014137574451} a^{13} - \frac{207719788076672791756570058890084814285}{1306587198137082201119588050014137574451} a^{11} + \frac{502414363853776085947223890243663936134}{1306587198137082201119588050014137574451} a^{9} + \frac{257617575451993487819026914554581468468}{1306587198137082201119588050014137574451} a^{7} + \frac{447529200623941735894547625894569861321}{1306587198137082201119588050014137574451} a^{5} - \frac{492116662567559112172956339576036663195}{1306587198137082201119588050014137574451} a^{3} - \frac{2530830192133113934360196053013280870}{12936506912248338624946416336773639351} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{2693200}$, which has order $43091200$ (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:  \( 38120.6275869 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1643:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 64 conjugacy class representatives for t16n1643 are not computed
Character table for t16n1643 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1

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 R $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
156506561Data not computed