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Properties

Label 20.0.16468295313...000.15
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $144.82$
Ramified primes $2, 5, 7, 11$
Class number $9063624$ (GRH)
Class group $[2, 4531812]$ (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![9456262613359, -5381490697486, 6534403195917, -2453356743028, 1493315910619, -365189713994, 187922132713, -30433556568, 15681102919, -1698262818, 879828741, -66394030, 34086743, -1779456, 905769, -32082, 16219, -356, 179, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 179*x^18 - 356*x^17 + 16219*x^16 - 32082*x^15 + 905769*x^14 - 1779456*x^13 + 34086743*x^12 - 66394030*x^11 + 879828741*x^10 - 1698262818*x^9 + 15681102919*x^8 - 30433556568*x^7 + 187922132713*x^6 - 365189713994*x^5 + 1493315910619*x^4 - 2453356743028*x^3 + 6534403195917*x^2 - 5381490697486*x + 9456262613359)
 
gp: K = bnfinit(x^20 - 2*x^19 + 179*x^18 - 356*x^17 + 16219*x^16 - 32082*x^15 + 905769*x^14 - 1779456*x^13 + 34086743*x^12 - 66394030*x^11 + 879828741*x^10 - 1698262818*x^9 + 15681102919*x^8 - 30433556568*x^7 + 187922132713*x^6 - 365189713994*x^5 + 1493315910619*x^4 - 2453356743028*x^3 + 6534403195917*x^2 - 5381490697486*x + 9456262613359, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 179 x^{18} - 356 x^{17} + 16219 x^{16} - 32082 x^{15} + 905769 x^{14} - 1779456 x^{13} + 34086743 x^{12} - 66394030 x^{11} + 879828741 x^{10} - 1698262818 x^{9} + 15681102919 x^{8} - 30433556568 x^{7} + 187922132713 x^{6} - 365189713994 x^{5} + 1493315910619 x^{4} - 2453356743028 x^{3} + 6534403195917 x^{2} - 5381490697486 x + 9456262613359 \)

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:  \(16468295313503070686136120314429440000000000=2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.82$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(69,·)$, $\chi_{3080}(321,·)$, $\chi_{3080}(1161,·)$, $\chi_{3080}(2829,·)$, $\chi_{3080}(589,·)$, $\chi_{3080}(2001,·)$, $\chi_{3080}(1429,·)$, $\chi_{3080}(601,·)$, $\chi_{3080}(29,·)$, $\chi_{3080}(1189,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(41,·)$, $\chi_{3080}(2029,·)$, $\chi_{3080}(2589,·)$, $\chi_{3080}(2549,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(2869,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{1}{11}$, $\frac{1}{11} a^{12} + \frac{1}{11} a$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{2}$, $\frac{1}{11} a^{14} + \frac{1}{11} a^{3}$, $\frac{1}{517} a^{15} + \frac{2}{47} a^{14} + \frac{3}{517} a^{13} + \frac{2}{517} a^{12} - \frac{21}{517} a^{11} - \frac{5}{517} a^{10} + \frac{49}{517} a^{9} + \frac{160}{517} a^{8} - \frac{160}{517} a^{7} - \frac{104}{517} a^{6} + \frac{170}{517} a^{5} + \frac{40}{517} a^{4} + \frac{247}{517} a^{3} - \frac{211}{517} a^{2} - \frac{257}{517} a - \frac{15}{517}$, $\frac{1}{110752239793723} a^{16} - \frac{86721694355}{110752239793723} a^{15} + \frac{1791602665451}{110752239793723} a^{14} - \frac{4748891271428}{110752239793723} a^{13} - \frac{530845487130}{110752239793723} a^{12} + \frac{4338661101017}{110752239793723} a^{11} + \frac{4367588235963}{110752239793723} a^{10} + \frac{11098557725633}{110752239793723} a^{9} - \frac{45432311700032}{110752239793723} a^{8} + \frac{47666361442847}{110752239793723} a^{7} + \frac{9500705947223}{110752239793723} a^{6} + \frac{12790437001974}{110752239793723} a^{5} + \frac{40158682558827}{110752239793723} a^{4} + \frac{38908625804992}{110752239793723} a^{3} + \frac{2646227104575}{10068385435793} a^{2} - \frac{4374500263813}{110752239793723} a - \frac{1707878907122}{110752239793723}$, $\frac{1}{110752239793723} a^{17} - \frac{8142717756}{110752239793723} a^{15} - \frac{2626946884877}{110752239793723} a^{14} - \frac{112366105567}{110752239793723} a^{13} + \frac{429833548422}{110752239793723} a^{12} - \frac{3577341156959}{110752239793723} a^{11} - \frac{938457559213}{110752239793723} a^{10} + \frac{13230502850750}{110752239793723} a^{9} - \frac{15898599285820}{110752239793723} a^{8} - \frac{9171569760014}{110752239793723} a^{7} - \frac{12915099780668}{110752239793723} a^{6} - \frac{10453840455236}{110752239793723} a^{5} - \frac{8679408008596}{110752239793723} a^{4} + \frac{24436331163532}{110752239793723} a^{3} + \frac{25989585375370}{110752239793723} a^{2} - \frac{4119783572519}{110752239793723} a - \frac{31691452372202}{110752239793723}$, $\frac{1}{110752239793723} a^{18} + \frac{84341297181}{110752239793723} a^{15} + \frac{4595398941858}{110752239793723} a^{14} - \frac{167152832229}{10068385435793} a^{13} + \frac{1253938187314}{110752239793723} a^{12} + \frac{2846810132407}{110752239793723} a^{11} + \frac{2423244108799}{110752239793723} a^{10} - \frac{2760198168296}{110752239793723} a^{9} + \frac{45439413573947}{110752239793723} a^{8} - \frac{24950214678594}{110752239793723} a^{7} + \frac{36962900467138}{110752239793723} a^{6} - \frac{13871075828519}{110752239793723} a^{5} - \frac{51253809393165}{110752239793723} a^{4} - \frac{33448330304783}{110752239793723} a^{3} + \frac{54335223435490}{110752239793723} a^{2} - \frac{35895296766569}{110752239793723} a + \frac{3776940990918}{110752239793723}$, $\frac{1}{963375662580693960168918334418563734356297266776675450137748114326683} a^{19} - \frac{2460842846231005852291731713709413630314111257674874230}{963375662580693960168918334418563734356297266776675450137748114326683} a^{18} - \frac{663662959712080045405176604002703876461236061493821806}{963375662580693960168918334418563734356297266776675450137748114326683} a^{17} - \frac{134938494394838527280763508614258099730406456077966961}{87579605689153996378992575856233066759663387888788677285249828575153} a^{16} - \frac{175042139461092582435983757714012395112555904550988709084455434955}{963375662580693960168918334418563734356297266776675450137748114326683} a^{15} + \frac{15960068405266364177482369763631987492978942597826258397961344346013}{963375662580693960168918334418563734356297266776675450137748114326683} a^{14} + \frac{33261663516019956160054955469341396108516311689585480560316783011415}{963375662580693960168918334418563734356297266776675450137748114326683} a^{13} + \frac{31894718588232727388763769555884173692378368356259386975742243160799}{963375662580693960168918334418563734356297266776675450137748114326683} a^{12} - \frac{27923229666818949687245779620586623794537775067275458195413800471364}{963375662580693960168918334418563734356297266776675450137748114326683} a^{11} - \frac{135340067201801524725429573007701464651677367939097503285395807926}{87579605689153996378992575856233066759663387888788677285249828575153} a^{10} - \frac{40432978144741490471443597368906501141528443190730895660835588936508}{87579605689153996378992575856233066759663387888788677285249828575153} a^{9} - \frac{236656095612596304418413423600655979844100132674491673735066400575164}{963375662580693960168918334418563734356297266776675450137748114326683} a^{8} + \frac{6434565701415662292601605449901939909160053608855346203714952493515}{20497354522993488514232304987629015624602069505886711705058470517589} a^{7} + \frac{399079357902460602964141551592225231383988315842814823142467101159731}{963375662580693960168918334418563734356297266776675450137748114326683} a^{6} + \frac{83182340972361397287421624930275004369287870211721428756976092493}{87579605689153996378992575856233066759663387888788677285249828575153} a^{5} + \frac{146903062269988271410940538982308817273811300401598483900750322364361}{963375662580693960168918334418563734356297266776675450137748114326683} a^{4} + \frac{470512022704042149519343234842799514409621006849392780752481819334094}{963375662580693960168918334418563734356297266776675450137748114326683} a^{3} + \frac{149635107101073242322942446551798497590518601062171731183529640555056}{963375662580693960168918334418563734356297266776675450137748114326683} a^{2} - \frac{235042847757715579036738020241514780169279907162070713786232729633042}{963375662580693960168918334418563734356297266776675450137748114326683} a - \frac{318803020563710023361929088066372988660944098452921207546564719339630}{963375662580693960168918334418563734356297266776675450137748114326683}$

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

Class group and class number

$C_{2}\times C_{4531812}$, which has order $9063624$ (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:  \( 1415140.1624943546 \) (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

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{-70}, \sqrt{77})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, 10.0.368919522607820800000.1, 10.0.241453843558400000.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed