↑

Properties

Label 20.4.11414008091...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{13}\cdot 13^{4}\cdot 41^{9}$
Root discriminant $25.29$
Ramified primes $5, 13, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1829, 10505, 13748, 4599, 4419, 13780, 8344, -8796, -9592, -1154, 3221, 736, 26, -336, 174, -156, 67, -30, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 22*x^18 - 30*x^17 + 67*x^16 - 156*x^15 + 174*x^14 - 336*x^13 + 26*x^12 + 736*x^11 + 3221*x^10 - 1154*x^9 - 9592*x^8 - 8796*x^7 + 8344*x^6 + 13780*x^5 + 4419*x^4 + 4599*x^3 + 13748*x^2 + 10505*x + 1829)
 
gp: K = bnfinit(x^20 - 8*x^19 + 22*x^18 - 30*x^17 + 67*x^16 - 156*x^15 + 174*x^14 - 336*x^13 + 26*x^12 + 736*x^11 + 3221*x^10 - 1154*x^9 - 9592*x^8 - 8796*x^7 + 8344*x^6 + 13780*x^5 + 4419*x^4 + 4599*x^3 + 13748*x^2 + 10505*x + 1829, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 22 x^{18} - 30 x^{17} + 67 x^{16} - 156 x^{15} + 174 x^{14} - 336 x^{13} + 26 x^{12} + 736 x^{11} + 3221 x^{10} - 1154 x^{9} - 9592 x^{8} - 8796 x^{7} + 8344 x^{6} + 13780 x^{5} + 4419 x^{4} + 4599 x^{3} + 13748 x^{2} + 10505 x + 1829 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11414008091096093897705078125=5^{13}\cdot 13^{4}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.29$
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:  $5, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{41} a^{16} - \frac{18}{41} a^{15} - \frac{20}{41} a^{14} - \frac{17}{41} a^{13} + \frac{5}{41} a^{12} - \frac{19}{41} a^{11} + \frac{1}{41} a^{10} - \frac{2}{41} a^{9} + \frac{9}{41} a^{8} - \frac{1}{41} a^{7} - \frac{11}{41} a^{6} - \frac{20}{41} a^{5} + \frac{17}{41} a^{3} + \frac{6}{41} a^{2} + \frac{16}{41} a - \frac{1}{41}$, $\frac{1}{41} a^{17} - \frac{16}{41} a^{15} - \frac{8}{41} a^{14} - \frac{14}{41} a^{13} - \frac{11}{41} a^{12} - \frac{13}{41} a^{11} + \frac{16}{41} a^{10} + \frac{14}{41} a^{9} - \frac{3}{41} a^{8} + \frac{12}{41} a^{7} - \frac{13}{41} a^{6} + \frac{9}{41} a^{5} + \frac{17}{41} a^{4} - \frac{16}{41} a^{3} + \frac{1}{41} a^{2} - \frac{18}{41}$, $\frac{1}{41} a^{18} - \frac{9}{41} a^{15} - \frac{6}{41} a^{14} + \frac{4}{41} a^{13} - \frac{15}{41} a^{12} - \frac{1}{41} a^{11} - \frac{11}{41} a^{10} + \frac{6}{41} a^{9} - \frac{8}{41} a^{8} + \frac{12}{41} a^{7} - \frac{3}{41} a^{6} - \frac{16}{41} a^{5} - \frac{16}{41} a^{4} - \frac{14}{41} a^{3} + \frac{14}{41} a^{2} - \frac{8}{41} a - \frac{16}{41}$, $\frac{1}{3624823444252478901006446648750675413} a^{19} + \frac{7984721658445387173957396289447302}{3624823444252478901006446648750675413} a^{18} - \frac{37700763286083358151311997785993823}{3624823444252478901006446648750675413} a^{17} + \frac{938066820137370850657416045866054}{88410327908597046366010893871967693} a^{16} + \frac{1414253915958644094283552498898463050}{3624823444252478901006446648750675413} a^{15} - \frac{1723643896230975393075488144411152582}{3624823444252478901006446648750675413} a^{14} + \frac{1155175235546484102249722105518451580}{3624823444252478901006446648750675413} a^{13} - \frac{167520068432019783374369050602691263}{3624823444252478901006446648750675413} a^{12} + \frac{781651200246418662154105913825900980}{3624823444252478901006446648750675413} a^{11} - \frac{151327549124509026112272142520063764}{3624823444252478901006446648750675413} a^{10} - \frac{697199154527842516299906799825281469}{3624823444252478901006446648750675413} a^{9} - \frac{236548148997892694648836644004581292}{3624823444252478901006446648750675413} a^{8} - \frac{384144449816523690978476485693073004}{3624823444252478901006446648750675413} a^{7} + \frac{1119897054863702076100819360084294006}{3624823444252478901006446648750675413} a^{6} - \frac{431228367610804601428178753653900719}{3624823444252478901006446648750675413} a^{5} - \frac{39252602655835708538877581319696741}{88410327908597046366010893871967693} a^{4} - \frac{741484234017395462986296960194992660}{3624823444252478901006446648750675413} a^{3} - \frac{356464321820467776787449469397706303}{3624823444252478901006446648750675413} a^{2} + \frac{106859603816930264876009487630902045}{3624823444252478901006446648750675413} a + \frac{1362127048829122819481319688123927546}{3624823444252478901006446648750675413}$

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:  $11$
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:  \( 1050250.97592 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.2665.1, 10.2.887778125.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$