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Properties

Label 20.0.12579538343...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{34}\cdot 43^{10}$
Root discriminant $101.15$
Ramified primes $5, 43$
Class number $2277632$ (GRH)
Class group $[4, 4, 4, 35588]$ (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![123003528649, -70408087320, 90457721015, -42585255340, 30304858095, -11980324785, 6117729630, -2056465185, 826391495, -237254320, 78129934, -19075585, 5230695, -1069245, 243935, -40330, 7515, -930, 135, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 135*x^18 - 930*x^17 + 7515*x^16 - 40330*x^15 + 243935*x^14 - 1069245*x^13 + 5230695*x^12 - 19075585*x^11 + 78129934*x^10 - 237254320*x^9 + 826391495*x^8 - 2056465185*x^7 + 6117729630*x^6 - 11980324785*x^5 + 30304858095*x^4 - 42585255340*x^3 + 90457721015*x^2 - 70408087320*x + 123003528649)
 
gp: K = bnfinit(x^20 - 10*x^19 + 135*x^18 - 930*x^17 + 7515*x^16 - 40330*x^15 + 243935*x^14 - 1069245*x^13 + 5230695*x^12 - 19075585*x^11 + 78129934*x^10 - 237254320*x^9 + 826391495*x^8 - 2056465185*x^7 + 6117729630*x^6 - 11980324785*x^5 + 30304858095*x^4 - 42585255340*x^3 + 90457721015*x^2 - 70408087320*x + 123003528649, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 135 x^{18} - 930 x^{17} + 7515 x^{16} - 40330 x^{15} + 243935 x^{14} - 1069245 x^{13} + 5230695 x^{12} - 19075585 x^{11} + 78129934 x^{10} - 237254320 x^{9} + 826391495 x^{8} - 2056465185 x^{7} + 6117729630 x^{6} - 11980324785 x^{5} + 30304858095 x^{4} - 42585255340 x^{3} + 90457721015 x^{2} - 70408087320 x + 123003528649 \)

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:  \(12579538343290477641858160495758056640625=5^{34}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.15$
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:  $5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1075=5^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1075}(1,·)$, $\chi_{1075}(386,·)$, $\chi_{1075}(259,·)$, $\chi_{1075}(644,·)$, $\chi_{1075}(646,·)$, $\chi_{1075}(1031,·)$, $\chi_{1075}(904,·)$, $\chi_{1075}(214,·)$, $\chi_{1075}(216,·)$, $\chi_{1075}(601,·)$, $\chi_{1075}(474,·)$, $\chi_{1075}(859,·)$, $\chi_{1075}(861,·)$, $\chi_{1075}(171,·)$, $\chi_{1075}(44,·)$, $\chi_{1075}(429,·)$, $\chi_{1075}(431,·)$, $\chi_{1075}(816,·)$, $\chi_{1075}(689,·)$, $\chi_{1075}(1074,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} - \frac{1}{7} a^{16} - \frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{16} - \frac{1}{7} a^{15} + \frac{3}{7} a^{14} + \frac{2}{7} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{19} + \frac{1876671191882444829283105355711679806230030734855800700070296241614955}{28196861055985029300305434189895174052578777297638364392282386951181157} a^{18} - \frac{1655697726294893701936254765645037433268453935344893828427322141126905268}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{17} - \frac{8160445330974935061706203043607689720095591629714952268176188204158157413}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{16} - \frac{616426305153740269416810195947260989704237010757831671649430620150091326}{6449024935804575987112714305431739094025517493359860198863443072691576051} a^{15} + \frac{1411106617249122409890563844704946421987860158976783461221181571789331263}{6449024935804575987112714305431739094025517493359860198863443072691576051} a^{14} + \frac{5631790299000724435530425132609070109560763852826265136860399014021769764}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{13} + \frac{1377617203402812647566348876573332092329820005959313187474347935288343381}{6449024935804575987112714305431739094025517493359860198863443072691576051} a^{12} - \frac{2825452194379235975540954197474346918492559053715148861019924656962319647}{6449024935804575987112714305431739094025517493359860198863443072691576051} a^{11} + \frac{5453148608986051445973249927025130460284691294494773498278805184591457562}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{10} + \frac{6396675565618902852481698297020427821993341211363458362178854207540639263}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{9} + \frac{16637081725542411065185452637238016609704763393178569134783660240690229685}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{8} - \frac{5157401250972795118191320506196929366866120637988737420796572545078721505}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{7} - \frac{5573135879394518787545409007208853235617273473397968833483847660313972118}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{6} - \frac{20995916006095726019758197132366891825811575333729298695141565700199577573}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{5} + \frac{16157343310585507823615598099228933136485133282547943692913535574964831742}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{4} - \frac{8906464557836971337085004808646232502419873134388979505397156216595338951}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{3} - \frac{3168672026278900697529003212332604044595811893422196991985126913688778553}{45143174550632031909789000138022173658178622453519021392044101508841032357} a^{2} - \frac{10379161736670264502563068713311829442185483970879376316751060020531323953}{45143174550632031909789000138022173658178622453519021392044101508841032357} a - \frac{1441865849987526669529358314240427822894029343673023453253067058531743863}{6449024935804575987112714305431739094025517493359860198863443072691576051}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{35588}$, which has order $2277632$ (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:  \( 161406.8376411007 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-215}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{5}, \sqrt{-43})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.112158541107177734375.3, 10.0.22431708221435546875.4

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$