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Properties

Label 20.2.32995339832...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{16}\cdot 5^{10}\cdot 19^{5}\cdot 461^{5}$
Root discriminant $37.66$
Ramified primes $2, 5, 19, 461$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8156, 22672, -42116, 65700, -60328, 33062, -5938, -5178, 8662, -7018, 3689, -1781, 1499, -328, -299, 93, 35, 10, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 9*x^18 + 10*x^17 + 35*x^16 + 93*x^15 - 299*x^14 - 328*x^13 + 1499*x^12 - 1781*x^11 + 3689*x^10 - 7018*x^9 + 8662*x^8 - 5178*x^7 - 5938*x^6 + 33062*x^5 - 60328*x^4 + 65700*x^3 - 42116*x^2 + 22672*x - 8156)
 
gp: K = bnfinit(x^20 - x^19 - 9*x^18 + 10*x^17 + 35*x^16 + 93*x^15 - 299*x^14 - 328*x^13 + 1499*x^12 - 1781*x^11 + 3689*x^10 - 7018*x^9 + 8662*x^8 - 5178*x^7 - 5938*x^6 + 33062*x^5 - 60328*x^4 + 65700*x^3 - 42116*x^2 + 22672*x - 8156, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 9 x^{18} + 10 x^{17} + 35 x^{16} + 93 x^{15} - 299 x^{14} - 328 x^{13} + 1499 x^{12} - 1781 x^{11} + 3689 x^{10} - 7018 x^{9} + 8662 x^{8} - 5178 x^{7} - 5938 x^{6} + 33062 x^{5} - 60328 x^{4} + 65700 x^{3} - 42116 x^{2} + 22672 x - 8156 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-32995339832393745791360000000000=-\,2^{16}\cdot 5^{10}\cdot 19^{5}\cdot 461^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.66$
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:  $2, 5, 19, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{68785932348066769280883363554139579024479365394} a^{19} - \frac{3577890980384434159098108725218225117800805049}{34392966174033384640441681777069789512239682697} a^{18} - \frac{3051442420997134738108030170064174614983736401}{68785932348066769280883363554139579024479365394} a^{17} + \frac{7714003353798360783571563985630951236036318924}{34392966174033384640441681777069789512239682697} a^{16} - \frac{4604726210927280680454525609455070705905615247}{68785932348066769280883363554139579024479365394} a^{15} - \frac{3003116660292060865763568190110696628611839827}{68785932348066769280883363554139579024479365394} a^{14} - \frac{9324844152510089268452003557184789184050512773}{34392966174033384640441681777069789512239682697} a^{13} + \frac{14592205812075466103593674421063553477714994527}{34392966174033384640441681777069789512239682697} a^{12} - \frac{24639718020426567533021660299110356645137798345}{68785932348066769280883363554139579024479365394} a^{11} + \frac{32854155391902507230054318094822733072280159303}{68785932348066769280883363554139579024479365394} a^{10} + \frac{424155320429195756542494087853581231243289212}{1810156114422809717917983251424725763802088563} a^{9} + \frac{10414695777959339279733959229937769188717695783}{34392966174033384640441681777069789512239682697} a^{8} - \frac{15082521524543597796838930480219975940332287598}{34392966174033384640441681777069789512239682697} a^{7} - \frac{4505161703214656187097469162045314735459204619}{68785932348066769280883363554139579024479365394} a^{6} - \frac{15684464706294477356466456559370110800741193255}{68785932348066769280883363554139579024479365394} a^{5} + \frac{36974350193759554448730168167940523029769949}{1810156114422809717917983251424725763802088563} a^{4} - \frac{14491482230791302354211807276744188981022654718}{34392966174033384640441681777069789512239682697} a^{3} + \frac{13227495330977465779748307870999880186137325462}{34392966174033384640441681777069789512239682697} a^{2} + \frac{3156554843656695469965999661746741689305442180}{34392966174033384640441681777069789512239682697} a + \frac{5900091087115304253823042830708206553467785133}{34392966174033384640441681777069789512239682697}$

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:  $10$
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:  \( 65207089.8888 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.61376064800000.1

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
461Data not computed