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Properties

Label 20.12.3828204809...0625.2
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 19^{6}\cdot 1699^{4}$
Root discriminant $23.94$
Ramified primes $5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T638

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -23, -77, 220, 210, -457, 292, 35, -606, 512, -121, -21, 217, -252, 120, -24, -31, 32, -6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 6*x^18 + 32*x^17 - 31*x^16 - 24*x^15 + 120*x^14 - 252*x^13 + 217*x^12 - 21*x^11 - 121*x^10 + 512*x^9 - 606*x^8 + 35*x^7 + 292*x^6 - 457*x^5 + 210*x^4 + 220*x^3 - 77*x^2 - 23*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 6*x^18 + 32*x^17 - 31*x^16 - 24*x^15 + 120*x^14 - 252*x^13 + 217*x^12 - 21*x^11 - 121*x^10 + 512*x^9 - 606*x^8 + 35*x^7 + 292*x^6 - 457*x^5 + 210*x^4 + 220*x^3 - 77*x^2 - 23*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 6 x^{18} + 32 x^{17} - 31 x^{16} - 24 x^{15} + 120 x^{14} - 252 x^{13} + 217 x^{12} - 21 x^{11} - 121 x^{10} + 512 x^{9} - 606 x^{8} + 35 x^{7} + 292 x^{6} - 457 x^{5} + 210 x^{4} + 220 x^{3} - 77 x^{2} - 23 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3828204809593746045712890625=5^{10}\cdot 19^{6}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.94$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{7}{19} a^{17} + \frac{8}{19} a^{16} + \frac{5}{19} a^{15} + \frac{8}{19} a^{14} + \frac{4}{19} a^{13} - \frac{3}{19} a^{12} + \frac{7}{19} a^{11} - \frac{1}{19} a^{10} - \frac{5}{19} a^{9} - \frac{1}{19} a^{8} + \frac{3}{19} a^{7} - \frac{7}{19} a^{6} + \frac{6}{19} a^{5} + \frac{3}{19} a^{4} - \frac{3}{19} a^{3} - \frac{7}{19} a^{2} - \frac{5}{19} a - \frac{1}{19}$, $\frac{1}{37976005736354304288109} a^{19} + \frac{743541841792458976196}{37976005736354304288109} a^{18} - \frac{10705489105524168358432}{37976005736354304288109} a^{17} + \frac{10969577901462095991154}{37976005736354304288109} a^{16} - \frac{16499999661740978819031}{37976005736354304288109} a^{15} + \frac{11696349783236607566178}{37976005736354304288109} a^{14} + \frac{12498624219251254940939}{37976005736354304288109} a^{13} + \frac{1032331143516543815420}{3452364157850391298919} a^{12} - \frac{704795028199600631743}{1998737144018647594111} a^{11} + \frac{826372617729327785475}{37976005736354304288109} a^{10} - \frac{3390787704581618743405}{37976005736354304288109} a^{9} + \frac{7830907136548492265325}{37976005736354304288109} a^{8} - \frac{12369865879670400490205}{37976005736354304288109} a^{7} + \frac{6000530652429103240161}{37976005736354304288109} a^{6} + \frac{18952782235282567026807}{37976005736354304288109} a^{5} - \frac{12284127039687970008429}{37976005736354304288109} a^{4} - \frac{13616371073908768141256}{37976005736354304288109} a^{3} - \frac{5523768250388284440556}{37976005736354304288109} a^{2} + \frac{7250136094073535105786}{37976005736354304288109} a + \frac{18846076490605290445283}{37976005736354304288109}$

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:  $15$
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:  \( 3627367.66857 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T638:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 65 conjugacy class representatives for t20n638 are not computed
Character table for t20n638 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$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}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1699Data not computed