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Properties

Label 20.4.15653924072...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{13}\cdot 1039^{4}\cdot 1049^{2}$
Root discriminant $22.89$
Ramified primes $5, 1039, 1049$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, -450, 2565, 9, -2086, -1504, -1369, 3217, 425, -66, -343, -697, 261, -12, 80, 57, -59, 7, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 + 7*x^17 - 59*x^16 + 57*x^15 + 80*x^14 - 12*x^13 + 261*x^12 - 697*x^11 - 343*x^10 - 66*x^9 + 425*x^8 + 3217*x^7 - 1369*x^6 - 1504*x^5 - 2086*x^4 + 9*x^3 + 2565*x^2 - 450*x + 89)
 
gp: K = bnfinit(x^20 - 3*x^19 + 3*x^18 + 7*x^17 - 59*x^16 + 57*x^15 + 80*x^14 - 12*x^13 + 261*x^12 - 697*x^11 - 343*x^10 - 66*x^9 + 425*x^8 + 3217*x^7 - 1369*x^6 - 1504*x^5 - 2086*x^4 + 9*x^3 + 2565*x^2 - 450*x + 89, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 3 x^{18} + 7 x^{17} - 59 x^{16} + 57 x^{15} + 80 x^{14} - 12 x^{13} + 261 x^{12} - 697 x^{11} - 343 x^{10} - 66 x^{9} + 425 x^{8} + 3217 x^{7} - 1369 x^{6} - 1504 x^{5} - 2086 x^{4} + 9 x^{3} + 2565 x^{2} - 450 x + 89 \)

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:  \(1565392407210041309814453125=5^{13}\cdot 1039^{4}\cdot 1049^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.89$
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:  $5, 1039, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{10} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{16} + \frac{1}{25} a^{15} + \frac{1}{25} a^{14} - \frac{1}{25} a^{13} - \frac{1}{25} a^{12} + \frac{2}{25} a^{11} - \frac{1}{25} a^{10} + \frac{4}{25} a^{9} - \frac{3}{25} a^{8} - \frac{4}{25} a^{7} + \frac{11}{25} a^{6} - \frac{9}{25} a^{5} + \frac{9}{25} a^{3} + \frac{2}{5} a^{2} - \frac{9}{25} a - \frac{11}{25}$, $\frac{1}{25} a^{17} - \frac{2}{25} a^{14} - \frac{2}{25} a^{12} - \frac{3}{25} a^{11} + \frac{1}{5} a^{10} + \frac{3}{25} a^{9} - \frac{1}{25} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{9}{25} a^{5} + \frac{4}{25} a^{4} + \frac{6}{25} a^{3} + \frac{11}{25} a^{2} - \frac{12}{25} a - \frac{9}{25}$, $\frac{1}{325} a^{18} + \frac{1}{325} a^{17} + \frac{1}{25} a^{15} - \frac{7}{325} a^{14} + \frac{3}{325} a^{13} - \frac{3}{65} a^{12} + \frac{37}{325} a^{11} + \frac{33}{325} a^{10} + \frac{47}{325} a^{9} + \frac{4}{325} a^{8} - \frac{16}{65} a^{7} - \frac{1}{325} a^{6} + \frac{133}{325} a^{5} - \frac{23}{65} a^{4} + \frac{17}{325} a^{3} - \frac{56}{325} a^{2} - \frac{61}{325} a - \frac{4}{325}$, $\frac{1}{2333717811035630316257675} a^{19} - \frac{3442746308796887931598}{2333717811035630316257675} a^{18} + \frac{13717901629510940294764}{2333717811035630316257675} a^{17} + \frac{2190785174952177687432}{179516754695048485865975} a^{16} - \frac{59606312243470498982561}{2333717811035630316257675} a^{15} + \frac{13226957874537151648116}{179516754695048485865975} a^{14} + \frac{139039728668450036294}{93348712441425212650307} a^{13} + \frac{50070246784855394325188}{2333717811035630316257675} a^{12} + \frac{659399818180975190311767}{2333717811035630316257675} a^{11} - \frac{28786529933311690143268}{2333717811035630316257675} a^{10} + \frac{17023897510643631198409}{179516754695048485865975} a^{9} - \frac{880544121913082966780718}{2333717811035630316257675} a^{8} + \frac{37953465851412643760719}{179516754695048485865975} a^{7} - \frac{10627000505032912515477}{42431232927920551204685} a^{6} - \frac{425196768110749357300792}{2333717811035630316257675} a^{5} + \frac{467912600677825422039144}{2333717811035630316257675} a^{4} - \frac{522962476897493086650514}{2333717811035630316257675} a^{3} + \frac{45712271161539700673511}{212156164639602756023425} a^{2} - \frac{280029372718143388868053}{2333717811035630316257675} a + \frac{78195403816545931073966}{2333717811035630316257675}$

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:  \( 324610.327643 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.4.3405971875.1

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 $20$ $20$ R $16{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$5.8.4.1$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}$
1039Data not computed
1049Data not computed