↑

Properties

Label 16.0.46803543212...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 61^{8}$
Root discriminant $26.12$
Ramified primes $5, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -1125, 775, 605, -2314, 1155, 2152, -5379, 5379, -16, -631, 92, 27, 23, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 + 23*x^13 + 27*x^12 + 92*x^11 - 631*x^10 - 16*x^9 + 5379*x^8 - 5379*x^7 + 2152*x^6 + 1155*x^5 - 2314*x^4 + 605*x^3 + 775*x^2 - 1125*x + 625)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 + 23*x^13 + 27*x^12 + 92*x^11 - 631*x^10 - 16*x^9 + 5379*x^8 - 5379*x^7 + 2152*x^6 + 1155*x^5 - 2314*x^4 + 605*x^3 + 775*x^2 - 1125*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} + 23 x^{13} + 27 x^{12} + 92 x^{11} - 631 x^{10} - 16 x^{9} + 5379 x^{8} - 5379 x^{7} + 2152 x^{6} + 1155 x^{5} - 2314 x^{4} + 605 x^{3} + 775 x^{2} - 1125 x + 625 \)

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:  \(46803543212226806640625=5^{12}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.12$
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:  $5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{6} + \frac{3}{10} a$, $\frac{1}{30} a^{12} - \frac{1}{30} a^{10} + \frac{1}{15} a^{9} + \frac{1}{3} a^{8} + \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{7}{30} a^{5} - \frac{2}{5} a^{4} + \frac{2}{15} a^{3} + \frac{13}{30} a^{2} + \frac{4}{15} a - \frac{1}{6}$, $\frac{1}{4333770150} a^{13} - \frac{823}{1828595} a^{12} - \frac{89184623}{2166885075} a^{11} + \frac{373949}{54857850} a^{10} - \frac{125685583}{2166885075} a^{9} + \frac{75796039}{288918010} a^{8} - \frac{311701166}{722295025} a^{7} - \frac{483642854}{2166885075} a^{6} + \frac{246643721}{1444590050} a^{5} - \frac{600957859}{2166885075} a^{4} - \frac{6776651}{54857850} a^{3} + \frac{610002946}{2166885075} a^{2} - \frac{166997}{5485785} a + \frac{17579801}{57783602}$, $\frac{1}{4333770150} a^{14} - \frac{50814661}{4333770150} a^{12} + \frac{1258072}{27428925} a^{11} - \frac{74715178}{2166885075} a^{10} + \frac{114881}{1992538} a^{9} + \frac{42605934}{722295025} a^{8} + \frac{2032912697}{4333770150} a^{7} - \frac{328749122}{722295025} a^{6} + \frac{563967056}{2166885075} a^{5} - \frac{6754421}{54857850} a^{4} - \frac{823855349}{2166885075} a^{3} - \frac{4873639}{10971570} a^{2} - \frac{53551626}{144459005} a - \frac{131321}{365719}$, $\frac{1}{21668850750} a^{15} - \frac{1}{10834425375} a^{14} - \frac{1}{21668850750} a^{13} - \frac{156446377}{21668850750} a^{12} - \frac{44168701}{7222950250} a^{11} - \frac{156031913}{3611475125} a^{10} + \frac{569222632}{10834425375} a^{9} + \frac{432917303}{7222950250} a^{8} - \frac{2282293801}{21668850750} a^{7} - \frac{7829009069}{21668850750} a^{6} + \frac{1324829246}{10834425375} a^{5} - \frac{16816622}{722295025} a^{4} - \frac{7263674759}{21668850750} a^{3} + \frac{508698703}{4333770150} a^{2} + \frac{289816733}{866754030} a + \frac{34318253}{173350806}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{3043353}{7222950250} a^{15} + \frac{223639}{249067250} a^{14} + \frac{9130059}{3611475125} a^{13} - \frac{69997119}{7222950250} a^{12} - \frac{82170531}{7222950250} a^{11} - \frac{139994238}{3611475125} a^{10} + \frac{2045409043}{7222950250} a^{9} + \frac{24346824}{3611475125} a^{8} - \frac{16370195787}{7222950250} a^{7} + \frac{16370195787}{7222950250} a^{6} - \frac{3274647828}{3611475125} a^{5} + \frac{2645530927}{1444590050} a^{4} + \frac{3521159421}{3611475125} a^{3} - \frac{368245713}{1444590050} a^{2} - \frac{94343943}{288918010} a + \frac{27390177}{57783602} \) (order $10$)
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:  \( 160289.989776 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{305}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 4.0.1525.1 x2, 4.0.18605.1 x2, 4.4.465125.1 x2, 4.4.7625.1 x2, 4.0.465125.1, \(\Q(\zeta_{5})\), 8.0.8653650625.2, 8.8.216341265625.1, 8.0.216341265625.2, 8.0.216341265625.1 x2, 8.0.58140625.2 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$