↑

Properties

Label 16.0.30825767705...7397.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{7}\cdot 53^{12}$
Root discriminant $60.33$
Ramified primes $13, 53$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1281

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![607, -5525, 24421, -52399, 74142, -92914, 100610, -82593, 52921, -27468, 11966, -4209, 1293, -301, 63, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 63*x^14 - 301*x^13 + 1293*x^12 - 4209*x^11 + 11966*x^10 - 27468*x^9 + 52921*x^8 - 82593*x^7 + 100610*x^6 - 92914*x^5 + 74142*x^4 - 52399*x^3 + 24421*x^2 - 5525*x + 607)
 
gp: K = bnfinit(x^16 - 8*x^15 + 63*x^14 - 301*x^13 + 1293*x^12 - 4209*x^11 + 11966*x^10 - 27468*x^9 + 52921*x^8 - 82593*x^7 + 100610*x^6 - 92914*x^5 + 74142*x^4 - 52399*x^3 + 24421*x^2 - 5525*x + 607, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 63 x^{14} - 301 x^{13} + 1293 x^{12} - 4209 x^{11} + 11966 x^{10} - 27468 x^{9} + 52921 x^{8} - 82593 x^{7} + 100610 x^{6} - 92914 x^{5} + 74142 x^{4} - 52399 x^{3} + 24421 x^{2} - 5525 x + 607 \)

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:  \(30825767705154553478835417397=13^{7}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.33$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{27} a^{10} + \frac{4}{27} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} + \frac{1}{27} a^{6} - \frac{11}{27} a^{4} - \frac{4}{9} a^{3} + \frac{1}{27} a^{2} + \frac{13}{27} a + \frac{4}{27}$, $\frac{1}{81} a^{11} - \frac{1}{81} a^{10} + \frac{4}{81} a^{9} + \frac{10}{27} a^{8} + \frac{34}{81} a^{7} + \frac{22}{81} a^{6} + \frac{16}{81} a^{5} + \frac{16}{81} a^{4} + \frac{34}{81} a^{3} - \frac{19}{81} a^{2} - \frac{7}{81} a + \frac{34}{81}$, $\frac{1}{66339} a^{12} - \frac{2}{22113} a^{11} + \frac{10}{819} a^{10} - \frac{3995}{66339} a^{9} + \frac{13357}{66339} a^{8} - \frac{29524}{66339} a^{7} + \frac{31001}{66339} a^{6} - \frac{6382}{66339} a^{5} - \frac{4621}{9477} a^{4} - \frac{24}{91} a^{3} - \frac{6725}{66339} a^{2} - \frac{5011}{22113} a - \frac{11465}{66339}$, $\frac{1}{199017} a^{13} + \frac{1}{199017} a^{12} + \frac{256}{66339} a^{11} + \frac{1675}{199017} a^{10} - \frac{14608}{199017} a^{9} + \frac{21325}{66339} a^{8} - \frac{42989}{199017} a^{7} - \frac{54731}{199017} a^{6} + \frac{7951}{28431} a^{5} - \frac{44908}{199017} a^{4} + \frac{69820}{199017} a^{3} + \frac{70570}{199017} a^{2} - \frac{50357}{199017} a - \frac{1988}{28431}$, $\frac{1}{23284989} a^{14} - \frac{1}{3326427} a^{13} + \frac{85}{23284989} a^{12} - \frac{419}{23284989} a^{11} - \frac{27007}{2587221} a^{10} + \frac{1218989}{23284989} a^{9} + \frac{594487}{3326427} a^{8} - \frac{678904}{23284989} a^{7} - \frac{4378132}{23284989} a^{6} + \frac{2997376}{7761663} a^{5} - \frac{99265}{862407} a^{4} - \frac{8977924}{23284989} a^{3} - \frac{9874054}{23284989} a^{2} - \frac{10824943}{23284989} a + \frac{9376000}{23284989}$, $\frac{1}{69854967} a^{15} + \frac{4}{7761663} a^{13} + \frac{176}{69854967} a^{12} - \frac{245996}{69854967} a^{11} - \frac{482452}{69854967} a^{10} - \frac{504317}{3326427} a^{9} + \frac{1721990}{23284989} a^{8} - \frac{9130460}{69854967} a^{7} + \frac{24915182}{69854967} a^{6} - \frac{3196742}{23284989} a^{5} - \frac{4454020}{69854967} a^{4} + \frac{20420434}{69854967} a^{3} - \frac{33373343}{69854967} a^{2} + \frac{8913785}{23284989} a - \frac{603281}{9979281}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 7055713.66898 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.3745777030801.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$