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Properties

Label 20.4.28550002061...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{31}\cdot 19^{10}$
Root discriminant $52.82$
Ramified primes $5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![293849141, -283432920, 520007480, 81011030, -334401910, -35696808, 95240190, -6471845, -12617985, 6783670, -1654471, -167595, 212270, -65095, 9665, 623, -835, 225, -10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 10*x^18 + 225*x^17 - 835*x^16 + 623*x^15 + 9665*x^14 - 65095*x^13 + 212270*x^12 - 167595*x^11 - 1654471*x^10 + 6783670*x^9 - 12617985*x^8 - 6471845*x^7 + 95240190*x^6 - 35696808*x^5 - 334401910*x^4 + 81011030*x^3 + 520007480*x^2 - 283432920*x + 293849141)
 
gp: K = bnfinit(x^20 - 5*x^19 - 10*x^18 + 225*x^17 - 835*x^16 + 623*x^15 + 9665*x^14 - 65095*x^13 + 212270*x^12 - 167595*x^11 - 1654471*x^10 + 6783670*x^9 - 12617985*x^8 - 6471845*x^7 + 95240190*x^6 - 35696808*x^5 - 334401910*x^4 + 81011030*x^3 + 520007480*x^2 - 283432920*x + 293849141, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 10 x^{18} + 225 x^{17} - 835 x^{16} + 623 x^{15} + 9665 x^{14} - 65095 x^{13} + 212270 x^{12} - 167595 x^{11} - 1654471 x^{10} + 6783670 x^{9} - 12617985 x^{8} - 6471845 x^{7} + 95240190 x^{6} - 35696808 x^{5} - 334401910 x^{4} + 81011030 x^{3} + 520007480 x^{2} - 283432920 x + 293849141 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28550002061766572296619415283203125=5^{31}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.82$
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:  $5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{1044} a^{18} - \frac{253}{1044} a^{17} + \frac{233}{1044} a^{16} + \frac{95}{522} a^{15} + \frac{23}{261} a^{14} + \frac{41}{1044} a^{13} + \frac{31}{116} a^{12} - \frac{44}{261} a^{11} - \frac{77}{348} a^{10} + \frac{427}{1044} a^{9} + \frac{2}{29} a^{8} + \frac{17}{1044} a^{7} - \frac{151}{1044} a^{6} + \frac{25}{522} a^{5} - \frac{257}{1044} a^{4} - \frac{4}{29} a^{3} - \frac{41}{116} a^{2} - \frac{64}{261} a - \frac{13}{36}$, $\frac{1}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{19} + \frac{115192929781377244644399119144737240563895309018831220983191511023942188884015039}{1801900217970726800417265972011576180467500362713197218770567481484016614856268881098} a^{18} - \frac{8269408953687276842845980906574484845430795343430908276664467617616674528594170073}{400422270660161511203836882669239151215000080602932715282348329218670358856948640244} a^{17} + \frac{678744516507642913486809046731903396055182028575694166965537629601657752103004282099}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{16} - \frac{144945365708024377961095857571444941503167644305840260912081026595975067526179870921}{1801900217970726800417265972011576180467500362713197218770567481484016614856268881098} a^{15} + \frac{149735363787077250954136388756610681583421447888873971497578202845837777084420930449}{2402533623960969067223021296015434907290000483617596291694089975312022153141691841464} a^{14} + \frac{73636811547365596174244191284867594543077810166899199896782823608325169591617335058}{900950108985363400208632986005788090233750181356598609385283740742008307428134440549} a^{13} - \frac{3220170356411172792225586736286347186373303391373731803733814785494364462752579306995}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{12} - \frac{2450617805108869142808780085556290634658071239977744291902486549858052480421969724293}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{11} - \frac{9676481077816120138395784501656568952755615565451934900107997962741837852869025535}{62134490274852648290250550759019868291982771128041283405881637292552297064009271762} a^{10} - \frac{282201624629061372356143742211969311041513523161211570127837891751880268190386085855}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{9} - \frac{2041573097516929613897303894157162114504453142808529947811216378111965047540008134953}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{8} + \frac{309231739928277556322709140765852022773484935824767399497455470936233388223941738299}{1201266811980484533611510648007717453645000241808798145847044987656011076570845920732} a^{7} - \frac{545410645103822120756017056541113550448752307885394107570408056685593740167842401677}{2402533623960969067223021296015434907290000483617596291694089975312022153141691841464} a^{6} + \frac{2995211015623459742496548442956501448635482553567125571140995911071844643526860736103}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{5} + \frac{1660884097938735691469040258495119354562226919235790505941267107665529402023214445763}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{4} - \frac{163046626687793205993554581866436075347977416191985393245940203893020211599074136399}{800844541320323022407673765338478302430000161205865430564696658437340717713897280488} a^{3} - \frac{2501657450200524272965757670917476256633248771869411659412345514457045690380327289365}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a^{2} + \frac{1127707130768695105763027689389108480719721155328194939393936607911550207241083766595}{7207600871882907201669063888046304721870001450852788875082269925936066459425075524392} a + \frac{43858678253835996224883738981553864301976529725756706668614991919785628522472813287}{248537961099410593161002203036079473167931084512165133623526549170209188256037087048}$

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:  $11$
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:  \( 2616130963.265845 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.45125.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$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}$
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}$