↑

Properties

Label 20.8.48775635110...1369.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{2}\cdot 71^{2}\cdot 401^{10}$
Root discriminant $34.23$
Ramified primes $3, 71, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -1647, 12105, -42948, 90530, -132369, 149165, -137160, 103943, -60421, 20608, 3374, -9972, 7100, -2940, 671, 23, -94, 43, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 43*x^18 - 94*x^17 + 23*x^16 + 671*x^15 - 2940*x^14 + 7100*x^13 - 9972*x^12 + 3374*x^11 + 20608*x^10 - 60421*x^9 + 103943*x^8 - 137160*x^7 + 149165*x^6 - 132369*x^5 + 90530*x^4 - 42948*x^3 + 12105*x^2 - 1647*x + 81)
 
gp: K = bnfinit(x^20 - 10*x^19 + 43*x^18 - 94*x^17 + 23*x^16 + 671*x^15 - 2940*x^14 + 7100*x^13 - 9972*x^12 + 3374*x^11 + 20608*x^10 - 60421*x^9 + 103943*x^8 - 137160*x^7 + 149165*x^6 - 132369*x^5 + 90530*x^4 - 42948*x^3 + 12105*x^2 - 1647*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 43 x^{18} - 94 x^{17} + 23 x^{16} + 671 x^{15} - 2940 x^{14} + 7100 x^{13} - 9972 x^{12} + 3374 x^{11} + 20608 x^{10} - 60421 x^{9} + 103943 x^{8} - 137160 x^{7} + 149165 x^{6} - 132369 x^{5} + 90530 x^{4} - 42948 x^{3} + 12105 x^{2} - 1647 x + 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4877563511063069089624758321369=3^{2}\cdot 71^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.23$
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:  $3, 71, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{4}{9} a^{9} + \frac{1}{3} a^{8} - \frac{2}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{4}{9} a^{10} + \frac{1}{3} a^{9} - \frac{2}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{4}{9} a^{10} - \frac{1}{9} a^{8} + \frac{2}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{16892230814596844937719053224183} a^{19} - \frac{11697702813905375268910822063}{544910671438607901216743652393} a^{18} + \frac{556059495678493835768463377437}{16892230814596844937719053224183} a^{17} - \frac{282020539074335053504716710260}{16892230814596844937719053224183} a^{16} + \frac{440557004376713307388124121077}{16892230814596844937719053224183} a^{15} - \frac{306331237054038763582389299068}{16892230814596844937719053224183} a^{14} + \frac{108952886193888802168971888890}{1876914534955204993079894802687} a^{13} + \frac{292441526662552977263340181895}{16892230814596844937719053224183} a^{12} + \frac{250917165006456281930576816588}{5630743604865614979239684408061} a^{11} + \frac{2600530877568928472267808756662}{16892230814596844937719053224183} a^{10} - \frac{97109797597915278553259290385}{16892230814596844937719053224183} a^{9} + \frac{2015623354246614216131849410058}{16892230814596844937719053224183} a^{8} + \frac{5716014824292860395452162509063}{16892230814596844937719053224183} a^{7} + \frac{2718788551348412902495628022221}{5630743604865614979239684408061} a^{6} - \frac{3670492461930226672817246580187}{16892230814596844937719053224183} a^{5} + \frac{60163305668755840298593248236}{1876914534955204993079894802687} a^{4} + \frac{7602825023542400777487341932625}{16892230814596844937719053224183} a^{3} + \frac{2687843134377101826393090840446}{5630743604865614979239684408061} a^{2} - \frac{133406881956494834403822792482}{625638178318401664359964934229} a + \frac{233887278898727540245559194745}{625638178318401664359964934229}$

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:  \( 83345443.6311 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed