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Properties

Label 20.0.47708645403...5625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{34}\cdot 31^{10}$
Root discriminant $85.89$
Ramified primes $5, 31$
Class number $711330$ (GRH)
Class group $[711330]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8253340801, -5632826490, 7621501460, -4259691715, 3244122870, -1518197925, 841255635, -334753815, 147612170, -50313265, 18339868, -5345635, 1632840, -401835, 102455, -20650, 4290, -660, 105, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 105*x^18 - 660*x^17 + 4290*x^16 - 20650*x^15 + 102455*x^14 - 401835*x^13 + 1632840*x^12 - 5345635*x^11 + 18339868*x^10 - 50313265*x^9 + 147612170*x^8 - 334753815*x^7 + 841255635*x^6 - 1518197925*x^5 + 3244122870*x^4 - 4259691715*x^3 + 7621501460*x^2 - 5632826490*x + 8253340801)
 
gp: K = bnfinit(x^20 - 10*x^19 + 105*x^18 - 660*x^17 + 4290*x^16 - 20650*x^15 + 102455*x^14 - 401835*x^13 + 1632840*x^12 - 5345635*x^11 + 18339868*x^10 - 50313265*x^9 + 147612170*x^8 - 334753815*x^7 + 841255635*x^6 - 1518197925*x^5 + 3244122870*x^4 - 4259691715*x^3 + 7621501460*x^2 - 5632826490*x + 8253340801, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 105 x^{18} - 660 x^{17} + 4290 x^{16} - 20650 x^{15} + 102455 x^{14} - 401835 x^{13} + 1632840 x^{12} - 5345635 x^{11} + 18339868 x^{10} - 50313265 x^{9} + 147612170 x^{8} - 334753815 x^{7} + 841255635 x^{6} - 1518197925 x^{5} + 3244122870 x^{4} - 4259691715 x^{3} + 7621501460 x^{2} - 5632826490 x + 8253340801 \)

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:  \(477086454036646173335611820220947265625=5^{34}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.89$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(775=5^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{775}(1,·)$, $\chi_{775}(774,·)$, $\chi_{775}(714,·)$, $\chi_{775}(526,·)$, $\chi_{775}(464,·)$, $\chi_{775}(466,·)$, $\chi_{775}(404,·)$, $\chi_{775}(216,·)$, $\chi_{775}(154,·)$, $\chi_{775}(156,·)$, $\chi_{775}(94,·)$, $\chi_{775}(681,·)$, $\chi_{775}(619,·)$, $\chi_{775}(621,·)$, $\chi_{775}(559,·)$, $\chi_{775}(371,·)$, $\chi_{775}(309,·)$, $\chi_{775}(311,·)$, $\chi_{775}(249,·)$, $\chi_{775}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{707} a^{18} + \frac{225}{707} a^{17} + \frac{335}{707} a^{16} + \frac{248}{707} a^{15} - \frac{313}{707} a^{14} + \frac{5}{101} a^{13} + \frac{32}{101} a^{12} - \frac{10}{101} a^{11} - \frac{239}{707} a^{10} - \frac{54}{707} a^{9} - \frac{334}{707} a^{7} - \frac{36}{101} a^{6} - \frac{17}{101} a^{5} - \frac{130}{707} a^{4} + \frac{321}{707} a^{3} + \frac{221}{707} a^{2} + \frac{246}{707} a + \frac{137}{707}$, $\frac{1}{5401499416828037139329082670598563475199266140200512844441284646743} a^{19} + \frac{506987225681274461975619919064309392359964988818143560135226929}{5401499416828037139329082670598563475199266140200512844441284646743} a^{18} - \frac{1089448255353642409968074433381755842546490301483844005418014860775}{5401499416828037139329082670598563475199266140200512844441284646743} a^{17} + \frac{380864407289549259493326663506291860901936701683588674730827959182}{771642773832576734189868952942651925028466591457216120634469235249} a^{16} + \frac{1680403889385828356269192404785566043580958139079672411373378586536}{5401499416828037139329082670598563475199266140200512844441284646743} a^{15} - \frac{2209387022765926192504476101649550632070986759347342724170069999593}{5401499416828037139329082670598563475199266140200512844441284646743} a^{14} + \frac{268055518587735361251682014237626052205099850375908475981813238615}{771642773832576734189868952942651925028466591457216120634469235249} a^{13} + \frac{284151207923153386807457715787514349966604280131715254268278956208}{771642773832576734189868952942651925028466591457216120634469235249} a^{12} - \frac{2069002635884900455065182949393124416700619186708606604165641400685}{5401499416828037139329082670598563475199266140200512844441284646743} a^{11} + \frac{2447702972699262935888826789785932254007558033961313754596200712443}{5401499416828037139329082670598563475199266140200512844441284646743} a^{10} - \frac{1177887390882529930821566266196942011459646188186919412236929792298}{5401499416828037139329082670598563475199266140200512844441284646743} a^{9} - \frac{1559861147852850739208568855448969974011574895050393071520886880982}{5401499416828037139329082670598563475199266140200512844441284646743} a^{8} - \frac{1167286232136912787449274436567572157791530698352523902699161120502}{5401499416828037139329082670598563475199266140200512844441284646743} a^{7} + \frac{259010433654242139557793062253943187736507480454200304829560193299}{771642773832576734189868952942651925028466591457216120634469235249} a^{6} - \frac{1039574319984400556608831044457880462197753295839118531600063775391}{5401499416828037139329082670598563475199266140200512844441284646743} a^{5} + \frac{1573785589191364803959800567598505615627707325847376973080681152007}{5401499416828037139329082670598563475199266140200512844441284646743} a^{4} - \frac{1020757733763249053220130130683428105020657088156863497287554327471}{5401499416828037139329082670598563475199266140200512844441284646743} a^{3} - \frac{781289092002239604471559187961845989553070646333771868899689958883}{5401499416828037139329082670598563475199266140200512844441284646743} a^{2} - \frac{76026824732864199770891763193580813418188773200087948971929348883}{771642773832576734189868952942651925028466591457216120634469235249} a + \frac{1048835202234782096766963159607769704581071658871884413529879026484}{5401499416828037139329082670598563475199266140200512844441284646743}$

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

Class group and class number

$C_{711330}$, which has order $711330$ (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:  \( 161406.837641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-155}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{5}, \sqrt{-31})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.21842308807373046875.3, 10.0.4368461761474609375.3

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

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
$31$31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$