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Properties

Label 20.0.44322362984...000.11
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}$
Root discriminant $340.67$
Ramified primes $2, 3, 5, 31$
Class number $500572160$ (GRH)
Class group $[2, 2, 4, 4, 4, 44, 44440]$ (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![1546528524025, 0, 716504608240, 0, 146200889141, 0, 16571144000, 0, 1100798474, 0, 36316000, 0, -17574, 0, -11840, 0, 3469, 0, -80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 80*x^18 + 3469*x^16 - 11840*x^14 - 17574*x^12 + 36316000*x^10 + 1100798474*x^8 + 16571144000*x^6 + 146200889141*x^4 + 716504608240*x^2 + 1546528524025)
 
gp: K = bnfinit(x^20 - 80*x^18 + 3469*x^16 - 11840*x^14 - 17574*x^12 + 36316000*x^10 + 1100798474*x^8 + 16571144000*x^6 + 146200889141*x^4 + 716504608240*x^2 + 1546528524025, 1)
 

Normalized defining polynomial

\( x^{20} - 80 x^{18} + 3469 x^{16} - 11840 x^{14} - 17574 x^{12} + 36316000 x^{10} + 1100798474 x^{8} + 16571144000 x^{6} + 146200889141 x^{4} + 716504608240 x^{2} + 1546528524025 \)

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:  \(443223629840246396758117587996045905756160000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $340.67$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(2371,·)$, $\chi_{3720}(1349,·)$, $\chi_{3720}(3719,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(2999,·)$, $\chi_{3720}(1709,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2131,·)$, $\chi_{3720}(2011,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(2851,·)$, $\chi_{3720}(869,·)$, $\chi_{3720}(3239,·)$, $\chi_{3720}(3629,·)$, $\chi_{3720}(91,·)$, $\chi_{3720}(1589,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(2879,·)$, $\chi_{3720}(959,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{4} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} - \frac{3}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} - \frac{3}{16} a$, $\frac{1}{80} a^{10} + \frac{1}{40} a^{6} + \frac{1}{80} a^{2}$, $\frac{1}{80} a^{11} + \frac{1}{40} a^{7} + \frac{1}{80} a^{3}$, $\frac{1}{320} a^{12} + \frac{7}{320} a^{8} + \frac{11}{320} a^{4} + \frac{1}{64}$, $\frac{1}{320} a^{13} + \frac{7}{320} a^{9} + \frac{11}{320} a^{5} + \frac{1}{64} a$, $\frac{1}{1600} a^{14} + \frac{3}{1600} a^{10} + \frac{3}{1600} a^{6} - \frac{1}{20} a^{4} + \frac{1}{1600} a^{2} - \frac{1}{20}$, $\frac{1}{1600} a^{15} + \frac{3}{1600} a^{11} + \frac{3}{1600} a^{7} - \frac{1}{20} a^{5} + \frac{1}{1600} a^{3} - \frac{1}{20} a$, $\frac{1}{26156800} a^{16} + \frac{1271}{6539200} a^{14} - \frac{9843}{6539200} a^{12} - \frac{10927}{6539200} a^{10} + \frac{26539}{13078400} a^{8} - \frac{39587}{6539200} a^{6} + \frac{508789}{6539200} a^{4} - \frac{734589}{6539200} a^{2} + \frac{1683901}{5231360}$, $\frac{1}{26156800} a^{17} + \frac{1271}{6539200} a^{15} - \frac{9843}{6539200} a^{13} - \frac{10927}{6539200} a^{11} + \frac{26539}{13078400} a^{9} - \frac{39587}{6539200} a^{7} + \frac{508789}{6539200} a^{5} - \frac{734589}{6539200} a^{3} + \frac{1683901}{5231360} a$, $\frac{1}{45730861251456593402144531635628800} a^{18} - \frac{3226418195850693250865489}{714544707054009271908508306806700} a^{16} + \frac{37123310409074376210470562507}{228654306257282967010722658178144} a^{14} - \frac{11868713666283924617094263253977}{11432715312864148350536132908907200} a^{12} - \frac{56155017061153552298428918234143}{22865430625728296701072265817814400} a^{10} - \frac{105610700074034197917081749469627}{11432715312864148350536132908907200} a^{8} - \frac{64788143709872253513570393637617}{714544707054009271908508306806700} a^{6} + \frac{523908972192944862555925862940661}{11432715312864148350536132908907200} a^{4} - \frac{18829152047155218315729763612260203}{45730861251456593402144531635628800} a^{2} + \frac{894315590044082455765419866761787}{2286543062572829670107226581781440}$, $\frac{1}{56870670398005162271939928819409797536000} a^{19} - \frac{10972347746419439720903923653757}{1137413407960103245438798576388195950720} a^{17} + \frac{2911053008247015504481338646862019211}{14217667599501290567984982204852449384000} a^{15} - \frac{1987645606130722182462721580934018043}{1421766759950129056798498220485244938400} a^{13} + \frac{143927745874835826321040960239465349863}{28435335199002581135969964409704898768000} a^{11} - \frac{10390770457054692399977795327939216391}{2843533519900258113596996440970489876800} a^{9} - \frac{195820955340711622235785442223915665769}{14217667599501290567984982204852449384000} a^{7} + \frac{10437003272715058363430274179854315273}{1421766759950129056798498220485244938400} a^{5} + \frac{13637114212281092213143869166717578003641}{56870670398005162271939928819409797536000} a^{3} + \frac{2190806705895962276507644305225298028087}{5687067039800516227193992881940979753600} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{44}\times C_{44440}$, which has order $500572160$ (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:  \( 243800068.8951959 \) (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

\(\Q(\sqrt{62}) \), \(\Q(\sqrt{-465}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-30}, \sqrt{62})\), 5.5.923521.1, 10.10.866373538960867328.1, 10.0.20559450192137769600000.1, 10.0.21222658262851891200000.1

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$