↑

Properties

Label 20.0.58299958569...0000.8
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $54.74$
Ramified primes $2, 5, 11$
Class number $8052$ (GRH)
Class group $[2, 4026]$ (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![8823079, -4511618, 8662447, -10442226, 11433163, -10101132, 8535738, -6099092, 3977799, -2260566, 1180345, -539374, 225633, -82468, 27546, -7868, 2029, -426, 79, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 79*x^18 - 426*x^17 + 2029*x^16 - 7868*x^15 + 27546*x^14 - 82468*x^13 + 225633*x^12 - 539374*x^11 + 1180345*x^10 - 2260566*x^9 + 3977799*x^8 - 6099092*x^7 + 8535738*x^6 - 10101132*x^5 + 11433163*x^4 - 10442226*x^3 + 8662447*x^2 - 4511618*x + 8823079)
 
gp: K = bnfinit(x^20 - 10*x^19 + 79*x^18 - 426*x^17 + 2029*x^16 - 7868*x^15 + 27546*x^14 - 82468*x^13 + 225633*x^12 - 539374*x^11 + 1180345*x^10 - 2260566*x^9 + 3977799*x^8 - 6099092*x^7 + 8535738*x^6 - 10101132*x^5 + 11433163*x^4 - 10442226*x^3 + 8662447*x^2 - 4511618*x + 8823079, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 79 x^{18} - 426 x^{17} + 2029 x^{16} - 7868 x^{15} + 27546 x^{14} - 82468 x^{13} + 225633 x^{12} - 539374 x^{11} + 1180345 x^{10} - 2260566 x^{9} + 3977799 x^{8} - 6099092 x^{7} + 8535738 x^{6} - 10101132 x^{5} + 11433163 x^{4} - 10442226 x^{3} + 8662447 x^{2} - 4511618 x + 8823079 \)

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:  \(58299958569124301174210560000000000=2^{30}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.74$
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:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(261,·)$, $\chi_{440}(201,·)$, $\chi_{440}(81,·)$, $\chi_{440}(381,·)$, $\chi_{440}(149,·)$, $\chi_{440}(89,·)$, $\chi_{440}(349,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(101,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(29,·)$, $\chi_{440}(109,·)$, $\chi_{440}(61,·)$, $\chi_{440}(49,·)$, $\chi_{440}(9,·)$, $\chi_{440}(189,·)$, $\chi_{440}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{8} a$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} - \frac{1}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{16} a$, $\frac{1}{14654680714290329687056} a^{18} - \frac{9}{14654680714290329687056} a^{17} + \frac{119259685823186288861}{7327340357145164843528} a^{16} - \frac{38159941942344705345}{7327340357145164843528} a^{15} - \frac{159065380949972688583}{3663670178572582421764} a^{14} - \frac{310997088960359635701}{7327340357145164843528} a^{13} + \frac{28982829541818033955}{1831835089286291210882} a^{12} - \frac{678696972503505656281}{7327340357145164843528} a^{11} + \frac{1551234899693740669529}{14654680714290329687056} a^{10} + \frac{1399641705625634096867}{14654680714290329687056} a^{9} - \frac{47933033844882731304}{915917544643145605441} a^{8} - \frac{572043290665876166413}{7327340357145164843528} a^{7} - \frac{453721059380391627413}{1831835089286291210882} a^{6} + \frac{367425788792435893641}{3663670178572582421764} a^{5} - \frac{617845601261807847297}{7327340357145164843528} a^{4} - \frac{136249453784638347337}{1831835089286291210882} a^{3} - \frac{3180496888456613596403}{14654680714290329687056} a^{2} + \frac{7016139992006329470477}{14654680714290329687056} a + \frac{37933258696869636003}{915917544643145605441}$, $\frac{1}{42243186950634018517006075088} a^{19} + \frac{1441277}{42243186950634018517006075088} a^{18} - \frac{1196912565562389242520243683}{42243186950634018517006075088} a^{17} + \frac{638417206085758290478826779}{21121593475317009258503037544} a^{16} - \frac{213842362646641395747991427}{10560796737658504629251518772} a^{15} - \frac{131690769342110619505244181}{2640199184414626157312879693} a^{14} - \frac{212931837621455408862995561}{21121593475317009258503037544} a^{13} + \frac{122760147702554404009207376}{2640199184414626157312879693} a^{12} + \frac{642649259522521510344612273}{42243186950634018517006075088} a^{11} - \frac{476400189543620128168378635}{42243186950634018517006075088} a^{10} - \frac{2266932993590587783529798135}{42243186950634018517006075088} a^{9} + \frac{1064303410156935535713135935}{10560796737658504629251518772} a^{8} + \frac{82062760210662881377415361}{2640199184414626157312879693} a^{7} + \frac{370703115414183724654866651}{5280398368829252314625759386} a^{6} + \frac{2356375244180300721016086631}{10560796737658504629251518772} a^{5} + \frac{2889492535061408288083075623}{21121593475317009258503037544} a^{4} - \frac{19778999769560764503529707779}{42243186950634018517006075088} a^{3} - \frac{14370451190126936817619357923}{42243186950634018517006075088} a^{2} + \frac{16466868819071055427553557035}{42243186950634018517006075088} a - \frac{5058290381693482937586979583}{10560796737658504629251518772}$

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

Class group and class number

$C_{2}\times C_{4026}$, which has order $8052$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.77265229938688.1, 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$