↑

Properties

Label 20.0.11569850838...7408.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 220873^{3}$
Root discriminant $17.91$
Ramified primes $2, 220873$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1022

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2213, 1738, 3238, 1966, 1948, 584, 1278, -1198, 2084, -2324, 2502, -2182, 1773, -1210, 768, -406, 196, -76, 26, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 26*x^18 - 76*x^17 + 196*x^16 - 406*x^15 + 768*x^14 - 1210*x^13 + 1773*x^12 - 2182*x^11 + 2502*x^10 - 2324*x^9 + 2084*x^8 - 1198*x^7 + 1278*x^6 + 584*x^5 + 1948*x^4 + 1966*x^3 + 3238*x^2 + 1738*x + 2213)
 
gp: K = bnfinit(x^20 - 6*x^19 + 26*x^18 - 76*x^17 + 196*x^16 - 406*x^15 + 768*x^14 - 1210*x^13 + 1773*x^12 - 2182*x^11 + 2502*x^10 - 2324*x^9 + 2084*x^8 - 1198*x^7 + 1278*x^6 + 584*x^5 + 1948*x^4 + 1966*x^3 + 3238*x^2 + 1738*x + 2213, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 26 x^{18} - 76 x^{17} + 196 x^{16} - 406 x^{15} + 768 x^{14} - 1210 x^{13} + 1773 x^{12} - 2182 x^{11} + 2502 x^{10} - 2324 x^{9} + 2084 x^{8} - 1198 x^{7} + 1278 x^{6} + 584 x^{5} + 1948 x^{4} + 1966 x^{3} + 3238 x^{2} + 1738 x + 2213 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11569850838123915570577408=2^{30}\cdot 220873^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.91$
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:  $2, 220873$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3189750508225935653364071262139} a^{19} + \frac{1492096555542669616672980847763}{3189750508225935653364071262139} a^{18} - \frac{1401754145538780723550141603692}{3189750508225935653364071262139} a^{17} - \frac{698104917141592019304761504915}{3189750508225935653364071262139} a^{16} + \frac{15719957538224998053268831524}{3189750508225935653364071262139} a^{15} - \frac{634808662396410420231557365679}{3189750508225935653364071262139} a^{14} - \frac{1217135362738186049339186124275}{3189750508225935653364071262139} a^{13} - \frac{1004362038792731396670570106473}{3189750508225935653364071262139} a^{12} + \frac{1148336768467158362224395319009}{3189750508225935653364071262139} a^{11} + \frac{682837458534111424849056786884}{3189750508225935653364071262139} a^{10} + \frac{267781693117691044290129669549}{3189750508225935653364071262139} a^{9} - \frac{1426828572260985207936670537782}{3189750508225935653364071262139} a^{8} - \frac{488374687547803600666788770650}{3189750508225935653364071262139} a^{7} - \frac{250982779498848177546699863387}{3189750508225935653364071262139} a^{6} - \frac{1437473899379346922468594526294}{3189750508225935653364071262139} a^{5} - \frac{176580277878974094239252453995}{3189750508225935653364071262139} a^{4} - \frac{554628011089459178141135585906}{3189750508225935653364071262139} a^{3} + \frac{324614188139774215655512927096}{3189750508225935653364071262139} a^{2} + \frac{195852311303030019932334536921}{3189750508225935653364071262139} a - \frac{41895145448930939683617466683}{3189750508225935653364071262139}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2977757894426293}{11180727103187302807} a^{19} - \frac{4602090413295997}{11180727103187302807} a^{18} + \frac{67335110558910698}{11180727103187302807} a^{17} - \frac{378883688507438362}{11180727103187302807} a^{16} + \frac{1108668404273819540}{11180727103187302807} a^{15} - \frac{2741266779776515843}{11180727103187302807} a^{14} + \frac{5188210174759662252}{11180727103187302807} a^{13} - \frac{9042513850613462603}{11180727103187302807} a^{12} + \frac{12651021639476124552}{11180727103187302807} a^{11} - \frac{16468617024378931356}{11180727103187302807} a^{10} + \frac{17363750310617095153}{11180727103187302807} a^{9} - \frac{17846421630113949182}{11180727103187302807} a^{8} + \frac{14195665207637853865}{11180727103187302807} a^{7} - \frac{13652625989488127878}{11180727103187302807} a^{6} + \frac{4497264915214365043}{11180727103187302807} a^{5} - \frac{18607192994505467326}{11180727103187302807} a^{4} - \frac{10595524139170745646}{11180727103187302807} a^{3} - \frac{30005365638453531161}{11180727103187302807} a^{2} - \frac{25151744339910639821}{11180727103187302807} a - \frac{26558970800596823427}{11180727103187302807} \) (order $4$)
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:  \( 21750.4745171 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.0.226173952.1

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 32 sibling: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
220873Data not computed