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Properties

Label 20.0.44400491855...2129.1
Degree $20$
Signature $[0, 10]$
Discriminant $23^{10}\cdot 41^{18}$
Root discriminant $135.63$
Ramified primes $23, 41$
Class number $5466120$ (GRH)
Class group $[5466120]$ (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![2887254431, -1824170168, 2471905719, -1187336885, 937985122, -377902854, 218588868, -76179063, 34877966, -10651910, 3997785, -1065569, 337750, -79515, 22149, -4578, 1139, -218, 51, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 51*x^18 - 218*x^17 + 1139*x^16 - 4578*x^15 + 22149*x^14 - 79515*x^13 + 337750*x^12 - 1065569*x^11 + 3997785*x^10 - 10651910*x^9 + 34877966*x^8 - 76179063*x^7 + 218588868*x^6 - 377902854*x^5 + 937985122*x^4 - 1187336885*x^3 + 2471905719*x^2 - 1824170168*x + 2887254431)
 
gp: K = bnfinit(x^20 - 8*x^19 + 51*x^18 - 218*x^17 + 1139*x^16 - 4578*x^15 + 22149*x^14 - 79515*x^13 + 337750*x^12 - 1065569*x^11 + 3997785*x^10 - 10651910*x^9 + 34877966*x^8 - 76179063*x^7 + 218588868*x^6 - 377902854*x^5 + 937985122*x^4 - 1187336885*x^3 + 2471905719*x^2 - 1824170168*x + 2887254431, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 51 x^{18} - 218 x^{17} + 1139 x^{16} - 4578 x^{15} + 22149 x^{14} - 79515 x^{13} + 337750 x^{12} - 1065569 x^{11} + 3997785 x^{10} - 10651910 x^{9} + 34877966 x^{8} - 76179063 x^{7} + 218588868 x^{6} - 377902854 x^{5} + 937985122 x^{4} - 1187336885 x^{3} + 2471905719 x^{2} - 1824170168 x + 2887254431 \)

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:  \(4440049185593367529770294835648190910892129=23^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.63$
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:  $23, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(943=23\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{943}(1,·)$, $\chi_{943}(898,·)$, $\chi_{943}(392,·)$, $\chi_{943}(139,·)$, $\chi_{943}(461,·)$, $\chi_{943}(206,·)$, $\chi_{943}(783,·)$, $\chi_{943}(277,·)$, $\chi_{943}(599,·)$, $\chi_{943}(344,·)$, $\chi_{943}(346,·)$, $\chi_{943}(666,·)$, $\chi_{943}(160,·)$, $\chi_{943}(737,·)$, $\chi_{943}(482,·)$, $\chi_{943}(804,·)$, $\chi_{943}(551,·)$, $\chi_{943}(45,·)$, $\chi_{943}(942,·)$, $\chi_{943}(597,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{747} a^{18} + \frac{31}{747} a^{17} - \frac{13}{249} a^{16} - \frac{31}{249} a^{15} + \frac{11}{249} a^{14} - \frac{4}{249} a^{13} - \frac{7}{249} a^{12} - \frac{2}{249} a^{11} + \frac{10}{747} a^{10} + \frac{118}{747} a^{9} - \frac{9}{83} a^{8} + \frac{25}{249} a^{7} + \frac{94}{249} a^{6} - \frac{92}{249} a^{5} + \frac{46}{249} a^{4} + \frac{110}{249} a^{3} + \frac{70}{747} a^{2} + \frac{292}{747} a - \frac{20}{249}$, $\frac{1}{41300173970734277561221976574169198900321104720923847303993678501} a^{19} + \frac{3110428605333974222370275754707849158616743847203678026437192}{13766724656911425853740658858056399633440368240307949101331226167} a^{18} - \frac{701280102426832815006376798816205101599149609518224230065931824}{41300173970734277561221976574169198900321104720923847303993678501} a^{17} - \frac{733279588598990582249135427481516820289724086083468642373261920}{41300173970734277561221976574169198900321104720923847303993678501} a^{16} + \frac{32341635652859687142015995898313307367438113009628054758204955}{4588908218970475284580219619352133211146789413435983033777075389} a^{15} - \frac{1698224869269817259492312227768086107003126994253087131983463007}{13766724656911425853740658858056399633440368240307949101331226167} a^{14} - \frac{98568587072887004234285109446069010764233187645982381239430983}{13766724656911425853740658858056399633440368240307949101331226167} a^{13} - \frac{1225072742838805415988201564280353749805163220432623622594099248}{13766724656911425853740658858056399633440368240307949101331226167} a^{12} + \frac{4837626605669510458852323893543788285274466439955677660734194539}{41300173970734277561221976574169198900321104720923847303993678501} a^{11} - \frac{43428694346353143122596486867349121802574565396606880035521383}{4588908218970475284580219619352133211146789413435983033777075389} a^{10} + \frac{5713016261364527121475069661222014968489759951120844134753174947}{41300173970734277561221976574169198900321104720923847303993678501} a^{9} + \frac{1685144187784254731624289967722777398718865564500878690830606969}{41300173970734277561221976574169198900321104720923847303993678501} a^{8} + \frac{1834758827745891838683407614268679463034672490542289003540167191}{4588908218970475284580219619352133211146789413435983033777075389} a^{7} + \frac{4586826092393749968760554053069215611508056216103254970807570471}{13766724656911425853740658858056399633440368240307949101331226167} a^{6} - \frac{1274352090217373960199032507607534269385541627360104476344695865}{13766724656911425853740658858056399633440368240307949101331226167} a^{5} - \frac{41878722945583954896491759416279288212579288814302141597737564}{13766724656911425853740658858056399633440368240307949101331226167} a^{4} + \frac{8862359451408781489077644540653853340586899277008039789731479363}{41300173970734277561221976574169198900321104720923847303993678501} a^{3} - \frac{3424928226120687398544430892749312868050213070223492319099497071}{13766724656911425853740658858056399633440368240307949101331226167} a^{2} - \frac{10407201539883894582152630806983080918520221409078263205447799555}{41300173970734277561221976574169198900321104720923847303993678501} a - \frac{7341426425428674449235217633807750467519554131393251805051554374}{41300173970734277561221976574169198900321104720923847303993678501}$

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

Class group and class number

$C_{5466120}$, which has order $5466120$ (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:  \( 5104264.636551031 \) (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

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-943}) \), \(\Q(\sqrt{-23}, \sqrt{41})\), 5.5.2825761.1, 10.10.327381934393961.1, 10.0.51393717603976344503.2, 10.0.2107142421763030124623.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$23$23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$41$41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$