↑

Properties

Label 20.0.26820899473...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{34}\cdot 41^{19}$
Root discriminant $1050.57$
Ramified primes $2, 5, 41$
Class number $31909519360$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 15580820]$ (GRH)
Galois group $C_{20}$ (as 20T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![66103908761600, 0, 116659093504000, 0, 66066967040000, 0, 14458216384000, 0, 1515651542800, 0, 81706435080, 0, 2274853225, 0, 33107500, 0, 243950, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 243950*x^16 + 33107500*x^14 + 2274853225*x^12 + 81706435080*x^10 + 1515651542800*x^8 + 14458216384000*x^6 + 66066967040000*x^4 + 116659093504000*x^2 + 66103908761600)
 
gp: K = bnfinit(x^20 + 820*x^18 + 243950*x^16 + 33107500*x^14 + 2274853225*x^12 + 81706435080*x^10 + 1515651542800*x^8 + 14458216384000*x^6 + 66066967040000*x^4 + 116659093504000*x^2 + 66103908761600, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 243950 x^{16} + 33107500 x^{14} + 2274853225 x^{12} + 81706435080 x^{10} + 1515651542800 x^{8} + 14458216384000 x^{6} + 66066967040000 x^{4} + 116659093504000 x^{2} + 66103908761600 \)

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:  \(2682089947307619115362844269019165039062500000000000000000000=2^{20}\cdot 5^{34}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1050.57$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4100=2^{2}\cdot 5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4100}(1,·)$, $\chi_{4100}(901,·)$, $\chi_{4100}(3979,·)$, $\chi_{4100}(461,·)$, $\chi_{4100}(1679,·)$, $\chi_{4100}(1619,·)$, $\chi_{4100}(2739,·)$, $\chi_{4100}(3599,·)$, $\chi_{4100}(3421,·)$, $\chi_{4100}(159,·)$, $\chi_{4100}(3739,·)$, $\chi_{4100}(3219,·)$, $\chi_{4100}(2341,·)$, $\chi_{4100}(681,·)$, $\chi_{4100}(1261,·)$, $\chi_{4100}(1841,·)$, $\chi_{4100}(3699,·)$, $\chi_{4100}(3859,·)$, $\chi_{4100}(2681,·)$, $\chi_{4100}(3221,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{8} - \frac{1}{32} a^{6} + \frac{1}{64} a^{4} + \frac{1}{16} a^{2}$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} + \frac{1}{128} a^{5} + \frac{1}{32} a^{3}$, $\frac{1}{3200} a^{10} + \frac{1}{320} a^{8} + \frac{1}{128} a^{6} - \frac{1}{8} a^{2} + \frac{1}{5}$, $\frac{1}{6400} a^{11} - \frac{3}{1280} a^{9} + \frac{3}{256} a^{7} + \frac{15}{256} a^{5} + \frac{15}{64} a^{3} + \frac{7}{20} a$, $\frac{1}{12800} a^{12} - \frac{1}{12800} a^{10} + \frac{3}{2560} a^{8} - \frac{19}{512} a^{6} - \frac{3}{128} a^{4} + \frac{7}{40} a^{2} + \frac{1}{5}$, $\frac{1}{1587200} a^{13} - \frac{37}{1587200} a^{11} + \frac{471}{317440} a^{9} - \frac{1487}{63488} a^{7} - \frac{97}{1984} a^{5} + \frac{4923}{19840} a^{3} + \frac{419}{1240} a$, $\frac{1}{73011200} a^{14} - \frac{57}{14602240} a^{12} - \frac{10293}{73011200} a^{10} - \frac{43891}{14602240} a^{8} - \frac{773}{15872} a^{6} + \frac{72193}{912640} a^{4} + \frac{304}{713} a^{2} + \frac{26}{115}$, $\frac{1}{73011200} a^{15} - \frac{9}{73011200} a^{13} + \frac{2311}{73011200} a^{11} + \frac{17657}{14602240} a^{9} + \frac{689}{31744} a^{7} + \frac{25503}{912640} a^{5} + \frac{16297}{228160} a^{3} - \frac{1057}{3565} a$, $\frac{1}{38257868800} a^{16} - \frac{1}{478223360} a^{14} - \frac{673207}{19128934400} a^{12} - \frac{49083}{478223360} a^{10} - \frac{3082563}{1530314752} a^{8} - \frac{19818809}{956446720} a^{6} - \frac{5310067}{95644672} a^{4} - \frac{12677487}{29888960} a^{2} + \frac{1428}{3013}$, $\frac{1}{306062950400} a^{17} - \frac{151}{76515737600} a^{15} - \frac{20041}{153031475200} a^{13} + \frac{1620131}{76515737600} a^{11} + \frac{50394773}{61212590080} a^{9} - \frac{32292629}{7651573760} a^{7} + \frac{150869253}{3825786880} a^{5} + \frac{37285389}{239111680} a^{3} - \frac{171111}{1868060} a$, $\frac{1}{169961975580906246146060386304000} a^{18} - \frac{61270234662726690499}{8498098779045312307303019315200} a^{16} + \frac{20624832841309832226603}{16996197558090624614606038630400} a^{14} - \frac{18104892181387653810450469}{1699619755809062461460603863040} a^{12} + \frac{3110950062228287847185194581}{33992395116181249229212077260800} a^{10} - \frac{12072386498524711199882829129}{4249049389522656153651509657600} a^{8} + \frac{1891044559836906554149938641}{424904938952265615365150965760} a^{6} - \frac{791381785443405358914839337}{26556558684516600960321935360} a^{4} + \frac{10001772520911340896578831}{165978491778228756002012096} a^{2} + \frac{135211241574264803251346}{418292570005616824601845}$, $\frac{1}{1359695804647249969168483090432000} a^{19} + \frac{49793175893029519153}{67984790232362498458424154521600} a^{17} + \frac{119249141414821346397579}{135969580464724996916848309043200} a^{15} - \frac{21065626072189447048310849}{67984790232362498458424154521600} a^{13} + \frac{17217355510896253360597917333}{271939160929449993833696618086400} a^{11} - \frac{110492157085595901673086892239}{33992395116181249229212077260800} a^{9} - \frac{11143611877252757174903726859}{679847902323624984584241545216} a^{7} - \frac{8639143672266303173930851}{829892458891143780010060480} a^{5} + \frac{1573576208600604937341293467}{13278279342258300480160967680} a^{3} - \frac{63771940977648580886922409}{207473114722785945002515120} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{15580820}$, which has order $31909519360$ (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:  \( 523576436698285.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.27568400.4, 5.5.1103812890625.4, 10.10.49954518797906646728515625.2

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 $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$5$5.10.17.26$x^{10} - 10 x^{8} + 85$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.26$x^{10} - 10 x^{8} + 85$$10$$1$$17$$C_{10}$$[2]_{2}$
41Data not computed