↑

Properties

Label 16.0.39930996351...3125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{7}\cdot 59^{10}$
Root discriminant $25.86$
Ramified primes $5, 59$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![81, -396, 606, -457, 578, -289, -190, -90, 279, -178, 113, -18, -12, -1, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 12*x^12 - 18*x^11 + 113*x^10 - 178*x^9 + 279*x^8 - 90*x^7 - 190*x^6 - 289*x^5 + 578*x^4 - 457*x^3 + 606*x^2 - 396*x + 81)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - x^13 - 12*x^12 - 18*x^11 + 113*x^10 - 178*x^9 + 279*x^8 - 90*x^7 - 190*x^6 - 289*x^5 + 578*x^4 - 457*x^3 + 606*x^2 - 396*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - x^{13} - 12 x^{12} - 18 x^{11} + 113 x^{10} - 178 x^{9} + 279 x^{8} - 90 x^{7} - 190 x^{6} - 289 x^{5} + 578 x^{4} - 457 x^{3} + 606 x^{2} - 396 x + 81 \)

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:  \(39930996351612609453125=5^{7}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.86$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{45} a^{14} - \frac{2}{15} a^{13} + \frac{2}{45} a^{12} - \frac{1}{15} a^{9} + \frac{4}{9} a^{8} - \frac{7}{15} a^{7} - \frac{4}{15} a^{6} - \frac{2}{5} a^{5} + \frac{1}{9} a^{4} + \frac{1}{15} a^{3} + \frac{11}{45} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{23022704005324155} a^{15} + \frac{6118614445979}{2558078222813795} a^{14} + \frac{89622963393031}{4604540801064831} a^{13} + \frac{329872151979661}{2558078222813795} a^{12} + \frac{21804027292676}{1534846933688277} a^{11} + \frac{326999140210904}{7674234668441385} a^{10} - \frac{3222883301852011}{23022704005324155} a^{9} - \frac{231886256307139}{2558078222813795} a^{8} + \frac{15331085009039}{2558078222813795} a^{7} + \frac{2942887117739161}{7674234668441385} a^{6} + \frac{39620328552929}{23022704005324155} a^{5} + \frac{1090353183796337}{2558078222813795} a^{4} - \frac{6363241788458443}{23022704005324155} a^{3} - \frac{1041182199292164}{2558078222813795} a^{2} + \frac{2460584559086276}{7674234668441385} a - \frac{982689252628238}{2558078222813795}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 98285.3111503 \) (assuming GRH)
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

\(\Q(\sqrt{-59}) \), 4.0.17405.1, 8.0.1514670125.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ R

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$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$59$59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$