↑

Properties

Label 16.12.5452166070...8125.1
Degree $16$
Signature $[12, 2]$
Discriminant $5^{8}\cdot 7^{8}\cdot 61\cdot 251^{4}$
Root discriminant $30.45$
Ramified primes $5, 7, 61, 251$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1806

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -234, -648, 1879, 589, -3017, -422, 3778, -1420, -1815, 1762, -382, -179, 106, -9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 9*x^14 + 106*x^13 - 179*x^12 - 382*x^11 + 1762*x^10 - 1815*x^9 - 1420*x^8 + 3778*x^7 - 422*x^6 - 3017*x^5 + 589*x^4 + 1879*x^3 - 648*x^2 - 234*x + 11)
 
gp: K = bnfinit(x^16 - 5*x^15 - 9*x^14 + 106*x^13 - 179*x^12 - 382*x^11 + 1762*x^10 - 1815*x^9 - 1420*x^8 + 3778*x^7 - 422*x^6 - 3017*x^5 + 589*x^4 + 1879*x^3 - 648*x^2 - 234*x + 11, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 9 x^{14} + 106 x^{13} - 179 x^{12} - 382 x^{11} + 1762 x^{10} - 1815 x^{9} - 1420 x^{8} + 3778 x^{7} - 422 x^{6} - 3017 x^{5} + 589 x^{4} + 1879 x^{3} - 648 x^{2} - 234 x + 11 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(545216607000444867578125=5^{8}\cdot 7^{8}\cdot 61\cdot 251^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.45$
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:  $5, 7, 61, 251$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{12} + \frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{2}{25} a^{9} + \frac{12}{25} a^{8} - \frac{2}{25} a^{7} + \frac{9}{25} a^{6} - \frac{9}{25} a^{5} + \frac{8}{25} a^{4} + \frac{11}{25} a^{3} + \frac{3}{25} a^{2} + \frac{7}{25} a + \frac{1}{25}$, $\frac{1}{50} a^{13} - \frac{1}{10} a^{11} - \frac{2}{25} a^{10} - \frac{3}{10} a^{9} - \frac{2}{25} a^{8} - \frac{7}{25} a^{7} + \frac{6}{25} a^{6} - \frac{9}{25} a^{5} - \frac{1}{25} a^{4} + \frac{6}{25} a^{3} + \frac{9}{50} a^{2} + \frac{7}{25} a - \frac{11}{50}$, $\frac{1}{250} a^{14} - \frac{1}{125} a^{13} - \frac{1}{250} a^{12} + \frac{7}{250} a^{10} - \frac{18}{125} a^{9} + \frac{1}{125} a^{8} - \frac{19}{125} a^{7} - \frac{28}{125} a^{6} - \frac{11}{125} a^{5} - \frac{61}{125} a^{4} + \frac{109}{250} a^{3} - \frac{21}{125} a^{2} + \frac{79}{250} a - \frac{37}{125}$, $\frac{1}{15250} a^{15} + \frac{9}{15250} a^{14} + \frac{117}{15250} a^{13} + \frac{219}{15250} a^{12} - \frac{163}{15250} a^{11} - \frac{1139}{15250} a^{10} + \frac{2058}{7625} a^{9} - \frac{3358}{7625} a^{8} - \frac{3497}{7625} a^{7} + \frac{3256}{7625} a^{6} - \frac{3427}{7625} a^{5} - \frac{4423}{15250} a^{4} - \frac{3383}{15250} a^{3} - \frac{2783}{15250} a^{2} - \frac{1517}{3050} a + \frac{163}{7625}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 2861030.42194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1806:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24576
The 88 conjugacy class representatives for t16n1806 are not computed
Character table for t16n1806 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.94540875625.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
61Data not computed
251Data not computed