Properties

Label 20.2.206...339.1
Degree $20$
Signature $[2, 9]$
Discriminant $-2.060\times 10^{47}$
Root discriminant $232.11$
Ramified prime $19$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 20T362

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888)
 
gp: K = bnfinit(x^20 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31141888, -538817408, -54940704, -588303992, 8889264, -245996344, 45895165, -48001353, 20865344, -5756183, 3851661, -787322, 325793, -112347, 15561, -8474, 1197, -247, 76, -5, 1]);
 

\( x^{20} - 5 x^{19} + 76 x^{18} - 247 x^{17} + 1197 x^{16} - 8474 x^{15} + 15561 x^{14} - 112347 x^{13} + 325793 x^{12} - 787322 x^{11} + 3851661 x^{10} - 5756183 x^{9} + 20865344 x^{8} - 48001353 x^{7} + 45895165 x^{6} - 245996344 x^{5} + 8889264 x^{4} - 588303992 x^{3} - 54940704 x^{2} - 538817408 x + 31141888 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-206007596521214410095208558252435839890349094339\)\(\medspace = -\,19^{37}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $232.11$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $19$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4}$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{9} - \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{32} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{13} + \frac{1}{32} a^{9} + \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} - \frac{3}{32} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{10} - \frac{1}{8} a^{7} + \frac{1}{32} a^{6} - \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{11} + \frac{1}{64} a^{10} - \frac{3}{64} a^{9} - \frac{1}{8} a^{8} + \frac{7}{64} a^{7} - \frac{1}{64} a^{6} - \frac{11}{64} a^{5} - \frac{1}{32} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{1}{64} a^{9} - \frac{3}{64} a^{8} + \frac{1}{32} a^{7} + \frac{3}{32} a^{6} + \frac{3}{64} a^{5} + \frac{1}{16} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{128} a^{17} - \frac{1}{128} a^{15} - \frac{1}{64} a^{13} + \frac{3}{128} a^{11} + \frac{3}{64} a^{9} - \frac{1}{128} a^{7} + \frac{1}{16} a^{6} + \frac{25}{128} a^{5} + \frac{1}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{512} a^{18} - \frac{1}{256} a^{17} + \frac{1}{512} a^{16} - \frac{1}{256} a^{15} + \frac{1}{256} a^{14} - \frac{1}{256} a^{13} - \frac{7}{512} a^{12} + \frac{5}{256} a^{11} - \frac{7}{256} a^{10} - \frac{7}{256} a^{9} + \frac{9}{512} a^{8} + \frac{29}{256} a^{7} + \frac{49}{512} a^{6} - \frac{5}{32} a^{5} + \frac{7}{64} a^{4} + \frac{9}{64} a^{3} + \frac{5}{16} a^{2}$, $\frac{1}{1066474769403131349909817646245942282985387136771725317498655232} a^{19} - \frac{327001297411492430773000872987840937471084301871628924711285}{1066474769403131349909817646245942282985387136771725317498655232} a^{18} - \frac{96765668110611742058045365085329471764581761321096930506271}{355491589801043783303272548748647427661795712257241772499551744} a^{17} - \frac{457073508586348366861875082056444259453072194213159606292789}{1066474769403131349909817646245942282985387136771725317498655232} a^{16} - \frac{1874941055747880561354776714534093998026394368032156907629737}{266618692350782837477454411561485570746346784192931329374663808} a^{15} - \frac{91109881256331467265801648049521705853929972147715345322793}{44436448725130472912909068593580928457724464032155221562443968} a^{14} - \frac{727270510696475805664024592062179540921987404327465405747667}{355491589801043783303272548748647427661795712257241772499551744} a^{13} - \frac{3239250353798595746850556241442815506400019069618688860023035}{355491589801043783303272548748647427661795712257241772499551744} a^{12} + \frac{3351165368661455915495612756926163442415241283477696446938111}{133309346175391418738727205780742785373173392096465664687331904} a^{11} + \frac{930697583471393946663355399505815845561176310830366530375633}{88872897450260945825818137187161856915448928064310443124887936} a^{10} - \frac{20580725715039412497030785901878117708429264363362093891252951}{355491589801043783303272548748647427661795712257241772499551744} a^{9} + \frac{6138884095615287307437208393849775481404352406583838085949263}{1066474769403131349909817646245942282985387136771725317498655232} a^{8} - \frac{42528155647583197091120077664555270511727207476051448195381379}{355491589801043783303272548748647427661795712257241772499551744} a^{7} + \frac{24316185183362434798017451336120028276709441749010478346804127}{355491589801043783303272548748647427661795712257241772499551744} a^{6} - \frac{32661042924148404781980135617215868435999782993378554550016615}{266618692350782837477454411561485570746346784192931329374663808} a^{5} - \frac{2213153332968011924028976074022931933907524913217140006126999}{11109112181282618228227267148395232114431116008038805390610992} a^{4} + \frac{1658706607018617403552292323702272511291318658008815619096119}{44436448725130472912909068593580928457724464032155221562443968} a^{3} + \frac{958459834359645579684637976429900972375911574840978569980505}{4165917067980981835585225180648212042911668503014552021479122} a^{2} + \frac{1758738090251872159348099188849946617109469136568922898614195}{8331834135961963671170450361296424085823337006029104042958244} a - \frac{180261549118181503756152399635571866947075775327783766413777}{2082958533990490917792612590324106021455834251507276010739561}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 496592090559000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{9}\cdot 496592090559000000 \cdot 1}{2\sqrt{206007596521214410095208558252435839890349094339}}\approx 33.3969642233715$ (assuming GRH)

Galois group

20T362:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 6840
The 21 conjugacy class representatives for t20n362
Character table for t20n362 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 40 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $20$ $18{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ $18{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $19{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $18{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
19Data not computed

Additional information

This field is associated with the 19-torsion points on any elliptic curve in the isogeny class 19.a.