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Properties

Label 20.0.88229434783...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{34}\cdot 13^{10}$
Root discriminant $222.47$
Ramified primes $2, 5, 13$
Class number $141961200$ (GRH)
Class group $[20, 7098060]$ (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![35635031225, 4529591500, 16896886500, 1091979500, 3926504425, 213099660, 607590850, 37357000, 64575575, 4179340, 5048464, 386320, 307790, 13800, 13030, 276, 565, -20, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^17 + 565*x^16 + 276*x^15 + 13030*x^14 + 13800*x^13 + 307790*x^12 + 386320*x^11 + 5048464*x^10 + 4179340*x^9 + 64575575*x^8 + 37357000*x^7 + 607590850*x^6 + 213099660*x^5 + 3926504425*x^4 + 1091979500*x^3 + 16896886500*x^2 + 4529591500*x + 35635031225)
 
gp: K = bnfinit(x^20 - 20*x^17 + 565*x^16 + 276*x^15 + 13030*x^14 + 13800*x^13 + 307790*x^12 + 386320*x^11 + 5048464*x^10 + 4179340*x^9 + 64575575*x^8 + 37357000*x^7 + 607590850*x^6 + 213099660*x^5 + 3926504425*x^4 + 1091979500*x^3 + 16896886500*x^2 + 4529591500*x + 35635031225, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{17} + 565 x^{16} + 276 x^{15} + 13030 x^{14} + 13800 x^{13} + 307790 x^{12} + 386320 x^{11} + 5048464 x^{10} + 4179340 x^{9} + 64575575 x^{8} + 37357000 x^{7} + 607590850 x^{6} + 213099660 x^{5} + 3926504425 x^{4} + 1091979500 x^{3} + 16896886500 x^{2} + 4529591500 x + 35635031225 \)

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:  \(88229434783360000000000000000000000000000000000=2^{40}\cdot 5^{34}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $222.47$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2600=2^{3}\cdot 5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2600}(1,·)$, $\chi_{2600}(1091,·)$, $\chi_{2600}(519,·)$, $\chi_{2600}(521,·)$, $\chi_{2600}(1611,·)$, $\chi_{2600}(1039,·)$, $\chi_{2600}(1041,·)$, $\chi_{2600}(2131,·)$, $\chi_{2600}(469,·)$, $\chi_{2600}(1559,·)$, $\chi_{2600}(1561,·)$, $\chi_{2600}(989,·)$, $\chi_{2600}(2079,·)$, $\chi_{2600}(2081,·)$, $\chi_{2600}(1509,·)$, $\chi_{2600}(2599,·)$, $\chi_{2600}(2029,·)$, $\chi_{2600}(51,·)$, $\chi_{2600}(2549,·)$, $\chi_{2600}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{25} a^{10} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{12}{25} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5}$, $\frac{1}{25} a^{11} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{12}{25} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a$, $\frac{1}{175} a^{12} + \frac{3}{175} a^{11} - \frac{3}{175} a^{10} - \frac{2}{35} a^{9} - \frac{17}{35} a^{8} - \frac{72}{175} a^{7} - \frac{31}{175} a^{6} - \frac{69}{175} a^{5} - \frac{9}{35} a^{4} + \frac{17}{35} a^{2} + \frac{16}{35} a - \frac{6}{35}$, $\frac{1}{175} a^{13} + \frac{2}{175} a^{11} - \frac{1}{175} a^{10} - \frac{11}{35} a^{9} + \frac{43}{175} a^{8} - \frac{12}{35} a^{7} + \frac{31}{175} a^{6} + \frac{22}{175} a^{5} - \frac{8}{35} a^{4} + \frac{17}{35} a^{3} + \frac{9}{35} a - \frac{17}{35}$, $\frac{1}{175} a^{14} + \frac{9}{25} a^{9} + \frac{8}{35} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{1575} a^{15} + \frac{1}{1575} a^{13} + \frac{1}{525} a^{12} + \frac{1}{63} a^{11} - \frac{1}{525} a^{10} - \frac{79}{315} a^{9} + \frac{116}{525} a^{8} + \frac{494}{1575} a^{7} - \frac{67}{315} a^{6} - \frac{493}{1575} a^{5} + \frac{11}{45} a^{4} - \frac{148}{315} a^{3} + \frac{86}{315} a^{2} - \frac{1}{21} a - \frac{1}{45}$, $\frac{1}{1575} a^{16} + \frac{1}{1575} a^{14} + \frac{1}{525} a^{13} - \frac{2}{1575} a^{12} - \frac{1}{75} a^{11} + \frac{1}{1575} a^{10} + \frac{206}{525} a^{9} + \frac{584}{1575} a^{8} - \frac{281}{1575} a^{7} - \frac{412}{1575} a^{6} - \frac{587}{1575} a^{5} + \frac{19}{63} a^{4} + \frac{86}{315} a^{3} + \frac{52}{105} a^{2} + \frac{2}{315} a - \frac{17}{35}$, $\frac{1}{1575} a^{17} + \frac{1}{525} a^{14} - \frac{1}{525} a^{13} + \frac{1}{525} a^{12} - \frac{2}{525} a^{11} - \frac{3}{175} a^{10} + \frac{709}{1575} a^{9} - \frac{719}{1575} a^{8} - \frac{1}{105} a^{7} - \frac{72}{175} a^{6} + \frac{239}{1575} a^{5} - \frac{1}{7} a^{4} - \frac{11}{315} a^{3} + \frac{4}{21} a^{2} - \frac{7}{15} a - \frac{29}{315}$, $\frac{1}{3636254326466879775} a^{18} - \frac{348093977362277}{1212084775488959925} a^{17} - \frac{296345042021447}{3636254326466879775} a^{16} + \frac{144102598790257}{1212084775488959925} a^{15} - \frac{1074162571880998}{727250865293375955} a^{14} + \frac{586099090050872}{1212084775488959925} a^{13} - \frac{288943420532639}{145450173058675191} a^{12} + \frac{1125254999119549}{242416955097791985} a^{11} + \frac{465047467933991}{145450173058675191} a^{10} - \frac{38784853997694268}{727250865293375955} a^{9} + \frac{775402459926477224}{3636254326466879775} a^{8} + \frac{283498980296687854}{3636254326466879775} a^{7} - \frac{818114966651315252}{3636254326466879775} a^{6} - \frac{89734355600951534}{3636254326466879775} a^{5} + \frac{114187754729170357}{242416955097791985} a^{4} + \frac{278489570783966006}{727250865293375955} a^{3} - \frac{7297190206351646}{80805651699263995} a^{2} + \frac{37557506214546508}{80805651699263995} a + \frac{64258221520337623}{242416955097791985}$, $\frac{1}{461989579706452302157936445895784132824602463975} a^{19} - \frac{13017751062629876554488306268}{461989579706452302157936445895784132824602463975} a^{18} - \frac{12878893699017536570902943995096821087356446}{65998511386636043165419492270826304689228923425} a^{17} + \frac{120016762541384467475770198434355946843912252}{461989579706452302157936445895784132824602463975} a^{16} + \frac{47165623173551460514656344584650440659064654}{153996526568817434052645481965261377608200821325} a^{15} - \frac{138423859265297222793993937975052855728592}{262942276440781048467806742114845835415254675} a^{14} + \frac{318556513830693941538439277307171679680902929}{153996526568817434052645481965261377608200821325} a^{13} + \frac{995942515795429053353113378355490421838318834}{461989579706452302157936445895784132824602463975} a^{12} - \frac{944746614643414963576998473582281562756827049}{65998511386636043165419492270826304689228923425} a^{11} - \frac{8490839615227328684765613041605276516191037108}{461989579706452302157936445895784132824602463975} a^{10} - \frac{200682947375869239233566793602645565849407529486}{461989579706452302157936445895784132824602463975} a^{9} - \frac{1812815629599113665572037473863087171999097849}{7333167931848449240602165807869589409914324825} a^{8} + \frac{1770232848429376097544448028660520026424914378}{92397915941290460431587289179156826564920492795} a^{7} + \frac{153741784286657431011570047293907885384738224913}{461989579706452302157936445895784132824602463975} a^{6} - \frac{64778843204566525144884112784680146673080413062}{461989579706452302157936445895784132824602463975} a^{5} + \frac{3580641412143834385795485917296570944666546658}{30799305313763486810529096393052275521640164265} a^{4} - \frac{513895100938665618028336248566078393125749016}{30799305313763486810529096393052275521640164265} a^{3} - \frac{1920369211199224496138776531624878892290847267}{18479583188258092086317457835831365312984098559} a^{2} + \frac{5545045966270354570316490864294011607403379849}{30799305313763486810529096393052275521640164265} a + \frac{1301691043405039177023464568022270427430320411}{13199702277327208633083898454165260937845784685}$

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

Class group and class number

$C_{20}\times C_{7098060}$, which has order $141961200$ (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:  \( 19344397.966990974 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{10}, \sqrt{-26})\), 5.5.390625.1, 10.10.25000000000000000.1, 10.0.1856465000000000000000.3, 10.0.290072656250000000000.3

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
$13$13.10.5.2$x^{10} - 57122 x^{2} + 2227758$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
13.10.5.2$x^{10} - 57122 x^{2} + 2227758$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$