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Properties

Label 20.0.20213777905...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 19^{5}\cdot 1699^{5}$
Root discriminant $51.91$
Ramified primes $3, 5, 19, 1699$
Class number $2232$ (GRH)
Class group $[2, 1116]$ (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![297859, -125820, 487862, -200424, 401105, -111821, 183155, -19136, 42851, 6616, 5375, 2371, 2070, -1037, 1271, -610, 337, -96, 34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 34*x^18 - 96*x^17 + 337*x^16 - 610*x^15 + 1271*x^14 - 1037*x^13 + 2070*x^12 + 2371*x^11 + 5375*x^10 + 6616*x^9 + 42851*x^8 - 19136*x^7 + 183155*x^6 - 111821*x^5 + 401105*x^4 - 200424*x^3 + 487862*x^2 - 125820*x + 297859)
 
gp: K = bnfinit(x^20 - 4*x^19 + 34*x^18 - 96*x^17 + 337*x^16 - 610*x^15 + 1271*x^14 - 1037*x^13 + 2070*x^12 + 2371*x^11 + 5375*x^10 + 6616*x^9 + 42851*x^8 - 19136*x^7 + 183155*x^6 - 111821*x^5 + 401105*x^4 - 200424*x^3 + 487862*x^2 - 125820*x + 297859, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 34 x^{18} - 96 x^{17} + 337 x^{16} - 610 x^{15} + 1271 x^{14} - 1037 x^{13} + 2070 x^{12} + 2371 x^{11} + 5375 x^{10} + 6616 x^{9} + 42851 x^{8} - 19136 x^{7} + 183155 x^{6} - 111821 x^{5} + 401105 x^{4} - 200424 x^{3} + 487862 x^{2} - 125820 x + 297859 \)

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:  \(20213777905110009806334605947265625=3^{10}\cdot 5^{10}\cdot 19^{5}\cdot 1699^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.91$
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:  $3, 5, 19, 1699$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is 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}{11699979130529178071396153205651840539796240557904833} a^{19} + \frac{5100030669219332937214154878882953029302590874751911}{11699979130529178071396153205651840539796240557904833} a^{18} - \frac{5424692906331882413938285252619834175495705035871398}{11699979130529178071396153205651840539796240557904833} a^{17} - \frac{832840737406060236260774802297338631572831551894630}{11699979130529178071396153205651840539796240557904833} a^{16} - \frac{5463959560381270627085947015897426796990260370668818}{11699979130529178071396153205651840539796240557904833} a^{15} + \frac{3546971752084865506471131741376449998341590489929475}{11699979130529178071396153205651840539796240557904833} a^{14} - \frac{5050087129563381180532508080193751923489645370239703}{11699979130529178071396153205651840539796240557904833} a^{13} - \frac{2651749488443387761971808683704202261451999047655876}{11699979130529178071396153205651840539796240557904833} a^{12} + \frac{2847658724084490175358427712800231280162582856382018}{11699979130529178071396153205651840539796240557904833} a^{11} + \frac{255751255779185444052977253734471175622397922154951}{615788375291009372178744905560623186305065292521307} a^{10} - \frac{3230857801581614965321058103469282897664621463098958}{11699979130529178071396153205651840539796240557904833} a^{9} - \frac{145462152572169477355159249749741242468511383004241}{11699979130529178071396153205651840539796240557904833} a^{8} + \frac{127487579744113358107973718814729470999307757209283}{615788375291009372178744905560623186305065292521307} a^{7} - \frac{1750726462525827926831964155105961679886516727586987}{11699979130529178071396153205651840539796240557904833} a^{6} + \frac{4220379708008631394653036646716522712328444890917121}{11699979130529178071396153205651840539796240557904833} a^{5} - \frac{5158772065782012644679556464891299574670535746137736}{11699979130529178071396153205651840539796240557904833} a^{4} - \frac{89532703607759976382408858021069885883295922470205}{11699979130529178071396153205651840539796240557904833} a^{3} + \frac{85345805744955855541105792776901821065080639785505}{615788375291009372178744905560623186305065292521307} a^{2} + \frac{1477778612259257902023093969163987304114774061062519}{11699979130529178071396153205651840539796240557904833} a - \frac{3989251025212205563866558660082844296035347162065865}{11699979130529178071396153205651840539796240557904833}$

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

Class group and class number

$C_{2}\times C_{1116}$, which has order $2232$ (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:  \( -1 \) (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:  \( 351148.02443 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.7263225.5, 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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$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$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$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}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
1699Data not computed