↑

Properties

Label 20.0.16210890375...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{34}\cdot 11^{10}$
Root discriminant $144.71$
Ramified primes $2, 5, 11$
Class number $6408744$ (GRH)
Class group $[6408744]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![456344399, -352395330, 1399553075, -262818330, 744271155, 66057816, 96564720, -20546060, 8465225, 10598860, -750365, 168920, 54730, -59420, 21815, -2402, 2340, -30, 80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 80*x^18 - 30*x^17 + 2340*x^16 - 2402*x^15 + 21815*x^14 - 59420*x^13 + 54730*x^12 + 168920*x^11 - 750365*x^10 + 10598860*x^9 + 8465225*x^8 - 20546060*x^7 + 96564720*x^6 + 66057816*x^5 + 744271155*x^4 - 262818330*x^3 + 1399553075*x^2 - 352395330*x + 456344399)
 
gp: K = bnfinit(x^20 + 80*x^18 - 30*x^17 + 2340*x^16 - 2402*x^15 + 21815*x^14 - 59420*x^13 + 54730*x^12 + 168920*x^11 - 750365*x^10 + 10598860*x^9 + 8465225*x^8 - 20546060*x^7 + 96564720*x^6 + 66057816*x^5 + 744271155*x^4 - 262818330*x^3 + 1399553075*x^2 - 352395330*x + 456344399, 1)
 

Normalized defining polynomial

\( x^{20} + 80 x^{18} - 30 x^{17} + 2340 x^{16} - 2402 x^{15} + 21815 x^{14} - 59420 x^{13} + 54730 x^{12} + 168920 x^{11} - 750365 x^{10} + 10598860 x^{9} + 8465225 x^{8} - 20546060 x^{7} + 96564720 x^{6} + 66057816 x^{5} + 744271155 x^{4} - 262818330 x^{3} + 1399553075 x^{2} - 352395330 x + 456344399 \)

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:  \(16210890375625000000000000000000000000000000=2^{30}\cdot 5^{34}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.71$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2200=2^{3}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2200}(1,·)$, $\chi_{2200}(1541,·)$, $\chi_{2200}(769,·)$, $\chi_{2200}(329,·)$, $\chi_{2200}(1869,·)$, $\chi_{2200}(1101,·)$, $\chi_{2200}(1429,·)$, $\chi_{2200}(1209,·)$, $\chi_{2200}(221,·)$, $\chi_{2200}(1761,·)$, $\chi_{2200}(549,·)$, $\chi_{2200}(881,·)$, $\chi_{2200}(1321,·)$, $\chi_{2200}(109,·)$, $\chi_{2200}(989,·)$, $\chi_{2200}(1649,·)$, $\chi_{2200}(2089,·)$, $\chi_{2200}(441,·)$, $\chi_{2200}(1981,·)$, $\chi_{2200}(661,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} - \frac{3}{49} a^{10} - \frac{1}{49} a^{8} + \frac{3}{49} a^{7} - \frac{3}{7} a^{6} + \frac{20}{49} a^{5} + \frac{3}{49} a^{4} + \frac{1}{7} a^{3} - \frac{20}{49} a^{2} - \frac{3}{49} a + \frac{2}{7}$, $\frac{1}{49} a^{12} - \frac{2}{49} a^{10} - \frac{1}{49} a^{9} + \frac{2}{49} a^{7} + \frac{6}{49} a^{6} + \frac{2}{7} a^{5} + \frac{9}{49} a^{4} + \frac{1}{49} a^{3} - \frac{2}{7} a^{2} - \frac{9}{49} a - \frac{1}{7}$, $\frac{1}{49} a^{13} - \frac{2}{49} a^{7} + \frac{3}{7} a^{6} + \frac{1}{49} a - \frac{3}{7}$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{49} a^{15} - \frac{2}{49} a^{9} + \frac{1}{49} a^{3}$, $\frac{1}{343} a^{16} + \frac{1}{343} a^{15} + \frac{2}{343} a^{14} - \frac{2}{343} a^{13} - \frac{2}{343} a^{12} + \frac{3}{343} a^{11} - \frac{2}{49} a^{10} - \frac{2}{49} a^{9} + \frac{23}{343} a^{7} + \frac{128}{343} a^{6} + \frac{130}{343} a^{5} - \frac{1}{343} a^{4} - \frac{113}{343} a^{3} - \frac{135}{343} a^{2} - \frac{1}{49} a$, $\frac{1}{343} a^{17} + \frac{1}{343} a^{15} + \frac{3}{343} a^{14} - \frac{2}{343} a^{12} - \frac{3}{343} a^{11} + \frac{3}{49} a^{10} + \frac{3}{49} a^{9} - \frac{5}{343} a^{8} - \frac{2}{49} a^{7} + \frac{9}{343} a^{6} + \frac{51}{343} a^{5} + \frac{23}{49} a^{4} + \frac{69}{343} a^{3} - \frac{47}{343} a^{2} - \frac{24}{49} a - \frac{2}{7}$, $\frac{1}{45214707633970301} a^{18} + \frac{43112505240927}{45214707633970301} a^{17} - \frac{48501357249671}{45214707633970301} a^{16} + \frac{393466645927132}{45214707633970301} a^{15} + \frac{333561723486949}{45214707633970301} a^{14} + \frac{243067505270206}{45214707633970301} a^{13} + \frac{73549574388256}{45214707633970301} a^{12} + \frac{198726224985684}{45214707633970301} a^{11} - \frac{83657163906663}{6459243947710043} a^{10} + \frac{2867053164576792}{45214707633970301} a^{9} + \frac{54715973307281}{45214707633970301} a^{8} - \frac{2704451185595209}{45214707633970301} a^{7} + \frac{18341994676105525}{45214707633970301} a^{6} + \frac{3126572560134150}{6459243947710043} a^{5} - \frac{3760234335008574}{45214707633970301} a^{4} + \frac{19368301082090167}{45214707633970301} a^{3} - \frac{19455814617636878}{45214707633970301} a^{2} + \frac{1073669584386590}{6459243947710043} a - \frac{82405627189427}{922749135387149}$, $\frac{1}{6353640498507467941402307189698984820331159068692403499268208443} a^{19} - \frac{55676082541979644514766576164705742211762187307}{6353640498507467941402307189698984820331159068692403499268208443} a^{18} + \frac{615877273942316584850408919754450619876888490842156126254456}{907662928358209705914615312814140688618737009813200499895458349} a^{17} - \frac{4204729528117353557394178949190905241649245509526991609708976}{6353640498507467941402307189698984820331159068692403499268208443} a^{16} - \frac{14544777684932216776362705984042221494540385144113926160255858}{6353640498507467941402307189698984820331159068692403499268208443} a^{15} + \frac{34175123727060782375326883288920991367065694828895057159089233}{6353640498507467941402307189698984820331159068692403499268208443} a^{14} - \frac{4401198233197856914371080902480449494153563180658936778742782}{907662928358209705914615312814140688618737009813200499895458349} a^{13} - \frac{31253802316047466802234841191064513786191822390143572954414016}{6353640498507467941402307189698984820331159068692403499268208443} a^{12} + \frac{30889205496624919057696575542881986903897135306926223094458865}{6353640498507467941402307189698984820331159068692403499268208443} a^{11} - \frac{374033823394408779399446412478640437887067107923422691257794857}{6353640498507467941402307189698984820331159068692403499268208443} a^{10} + \frac{97716545956536843996823681114648058842820157414964842397268800}{6353640498507467941402307189698984820331159068692403499268208443} a^{9} - \frac{15429479453254291608584264132783030469708507385885596220639004}{907662928358209705914615312814140688618737009813200499895458349} a^{8} + \frac{19587595761058108341904468049608618213628912927476229258905577}{6353640498507467941402307189698984820331159068692403499268208443} a^{7} - \frac{1414965209186331931554781262817512335887133112374266998745745663}{6353640498507467941402307189698984820331159068692403499268208443} a^{6} + \frac{529478255375258318518599359684460998530353507326713238632356402}{6353640498507467941402307189698984820331159068692403499268208443} a^{5} - \frac{337760581718949978099670527402967262759309900971478362733070958}{907662928358209705914615312814140688618737009813200499895458349} a^{4} - \frac{768651491303465953778820879279474135653042710700579373478913289}{6353640498507467941402307189698984820331159068692403499268208443} a^{3} - \frac{3086656012322869949564268716890380552607768103278193715454920323}{6353640498507467941402307189698984820331159068692403499268208443} a^{2} - \frac{19759918028467255007039788304308553829935283595256135213047306}{907662928358209705914615312814140688618737009813200499895458349} a + \frac{27884126186887752255448449217266572905424708240092130985183382}{129666132622601386559230758973448669802676715687600071413636907}$

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

Class group and class number

$C_{6408744}$, which has order $6408744$ (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:  \( 42294001.73672045 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{2}, \sqrt{-55})\), 5.5.390625.1, 10.10.5000000000000000.1, 10.0.122872161865234375.1, 10.0.4026275000000000000000.3

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
5Data not computed
11Data not computed