↑

Properties

Label 20.14.1467214799...9375.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,3^{10}\cdot 5^{10}\cdot 239^{9}$
Root discriminant $45.53$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -44, 124, 7338, -12502, -19900, 42749, 8426, -45780, 1662, 34205, -11190, -12751, 8148, 885, -1695, 189, 139, -28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 28*x^18 + 139*x^17 + 189*x^16 - 1695*x^15 + 885*x^14 + 8148*x^13 - 12751*x^12 - 11190*x^11 + 34205*x^10 + 1662*x^9 - 45780*x^8 + 8426*x^7 + 42749*x^6 - 19900*x^5 - 12502*x^4 + 7338*x^3 + 124*x^2 - 44*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 28*x^18 + 139*x^17 + 189*x^16 - 1695*x^15 + 885*x^14 + 8148*x^13 - 12751*x^12 - 11190*x^11 + 34205*x^10 + 1662*x^9 - 45780*x^8 + 8426*x^7 + 42749*x^6 - 19900*x^5 - 12502*x^4 + 7338*x^3 + 124*x^2 - 44*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 28 x^{18} + 139 x^{17} + 189 x^{16} - 1695 x^{15} + 885 x^{14} + 8148 x^{13} - 12751 x^{12} - 11190 x^{11} + 34205 x^{10} + 1662 x^{9} - 45780 x^{8} + 8426 x^{7} + 42749 x^{6} - 19900 x^{5} - 12502 x^{4} + 7338 x^{3} + 124 x^{2} - 44 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1467214799841538367562738193359375=-\,3^{10}\cdot 5^{10}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{141} a^{17} + \frac{3}{47} a^{16} + \frac{6}{47} a^{15} - \frac{26}{141} a^{14} + \frac{28}{141} a^{13} - \frac{53}{141} a^{12} + \frac{52}{141} a^{11} - \frac{19}{47} a^{10} + \frac{35}{141} a^{9} + \frac{58}{141} a^{8} + \frac{41}{141} a^{7} - \frac{56}{141} a^{6} - \frac{14}{141} a^{5} + \frac{64}{141} a^{4} - \frac{26}{141} a^{3} + \frac{11}{141} a^{2} + \frac{53}{141} a + \frac{19}{141}$, $\frac{1}{2679} a^{18} + \frac{2}{2679} a^{17} + \frac{49}{2679} a^{16} - \frac{129}{893} a^{15} - \frac{118}{893} a^{14} - \frac{177}{893} a^{13} + \frac{25}{57} a^{12} + \frac{173}{893} a^{11} - \frac{788}{2679} a^{10} - \frac{407}{893} a^{9} + \frac{411}{893} a^{8} + \frac{926}{2679} a^{7} + \frac{2}{2679} a^{6} + \frac{11}{141} a^{5} - \frac{299}{893} a^{4} + \frac{710}{2679} a^{3} + \frac{415}{893} a^{2} - \frac{321}{893} a - \frac{368}{2679}$, $\frac{1}{19848031393388003453285912307} a^{19} - \frac{742818934058008904730905}{6616010464462667817761970769} a^{18} + \frac{19874570646735887703558706}{19848031393388003453285912307} a^{17} - \frac{1542967344393835030150085098}{19848031393388003453285912307} a^{16} + \frac{11009787564411615040261360}{348211077076982516724314251} a^{15} + \frac{3929176292053851269030224013}{19848031393388003453285912307} a^{14} - \frac{1518096858928978457139831136}{19848031393388003453285912307} a^{13} + \frac{1651283365762188186833827235}{6616010464462667817761970769} a^{12} + \frac{8085183584813841940145267218}{19848031393388003453285912307} a^{11} + \frac{1890771363852501397838552227}{6616010464462667817761970769} a^{10} - \frac{837264416900871083235859139}{1804366490308000313935082937} a^{9} + \frac{3897219424838316728719788892}{19848031393388003453285912307} a^{8} - \frac{1520162851967691717609101866}{6616010464462667817761970769} a^{7} + \frac{9854323336694321101929976666}{19848031393388003453285912307} a^{6} - \frac{2660023892419360374175205144}{19848031393388003453285912307} a^{5} - \frac{436003482953729416660028333}{6616010464462667817761970769} a^{4} + \frac{6158850695434732415553220354}{19848031393388003453285912307} a^{3} - \frac{1706585424341601392352526650}{6616010464462667817761970769} a^{2} - \frac{872014203465477823753083109}{6616010464462667817761970769} a + \frac{1697876540520187841872749353}{6616010464462667817761970769}$

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:  $16$
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:  \( 4882434911.31 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed