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Properties

Label 20.8.72797244630...8125.2
Degree $20$
Signature $[8, 6]$
Discriminant $5^{14}\cdot 19^{4}\cdot 29^{6}\cdot 109^{5}$
Root discriminant $49.33$
Ramified primes $5, 19, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 97, 588, 1743, 2943, 618, -4844, -1869, 3115, -1324, -368, 1621, -604, -245, 226, -40, -19, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^18 - 19*x^17 - 40*x^16 + 226*x^15 - 245*x^14 - 604*x^13 + 1621*x^12 - 368*x^11 - 1324*x^10 + 3115*x^9 - 1869*x^8 - 4844*x^7 + 618*x^6 + 2943*x^5 + 1743*x^4 + 588*x^3 + 97*x^2 - x - 1)
 
gp: K = bnfinit(x^20 - x^18 - 19*x^17 - 40*x^16 + 226*x^15 - 245*x^14 - 604*x^13 + 1621*x^12 - 368*x^11 - 1324*x^10 + 3115*x^9 - 1869*x^8 - 4844*x^7 + 618*x^6 + 2943*x^5 + 1743*x^4 + 588*x^3 + 97*x^2 - x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{18} - 19 x^{17} - 40 x^{16} + 226 x^{15} - 245 x^{14} - 604 x^{13} + 1621 x^{12} - 368 x^{11} - 1324 x^{10} + 3115 x^{9} - 1869 x^{8} - 4844 x^{7} + 618 x^{6} + 2943 x^{5} + 1743 x^{4} + 588 x^{3} + 97 x^{2} - x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7279724463006629630160577392578125=5^{14}\cdot 19^{4}\cdot 29^{6}\cdot 109^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.33$
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:  $5, 19, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{87401120753711682943824335259562} a^{19} + \frac{22152590897828785502910063544727}{87401120753711682943824335259562} a^{18} + \frac{12751971247264156246290695501398}{43700560376855841471912167629781} a^{17} + \frac{38825896857713542167588224060175}{87401120753711682943824335259562} a^{16} + \frac{3568707461157071243467703519607}{87401120753711682943824335259562} a^{15} - \frac{7712934838166805484164683903867}{87401120753711682943824335259562} a^{14} + \frac{18746415917248361568025163021016}{43700560376855841471912167629781} a^{13} - \frac{160097310957169659035924596956}{891848170956241662692085053669} a^{12} - \frac{18701854429137641619530515908915}{87401120753711682943824335259562} a^{11} - \frac{42225108616390269069331749270103}{87401120753711682943824335259562} a^{10} + \frac{14359140921578209921723961233277}{87401120753711682943824335259562} a^{9} - \frac{14145579632124739836573572065005}{43700560376855841471912167629781} a^{8} - \frac{22180646870126986984374569533929}{87401120753711682943824335259562} a^{7} - \frac{41637897395978730525418034644617}{87401120753711682943824335259562} a^{6} + \frac{33925291724588389640442570055533}{87401120753711682943824335259562} a^{5} + \frac{1559200507291629302140918906822}{6242937196693691638844595375683} a^{4} + \frac{26233656541420022418670299219}{61506770410775287082212762322} a^{3} + \frac{128533605777994147088091301567}{1783696341912483325384170107338} a^{2} - \frac{3121827604107038322064898470747}{43700560376855841471912167629781} a - \frac{16132250812210743613125248637305}{87401120753711682943824335259562}$

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:  $13$
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:  \( 4386602955.14 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8172298511640625.1

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 $16{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ $20$ $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R $20$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{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}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.4.3.3$x^{4} + 654$$4$$1$$3$$C_4$$[\ ]_{4}$