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Properties

Label 20.0.95797910278...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 19^{10}$
Root discriminant $44.57$
Ramified primes $2, 5, 19$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1267924, -205100, 2809180, -588360, 4621740, -105998, 4242720, 96710, 2183600, 36060, 681153, 3810, 134325, 70, 16820, -4, 1290, 0, 55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 55*x^18 + 1290*x^16 - 4*x^15 + 16820*x^14 + 70*x^13 + 134325*x^12 + 3810*x^11 + 681153*x^10 + 36060*x^9 + 2183600*x^8 + 96710*x^7 + 4242720*x^6 - 105998*x^5 + 4621740*x^4 - 588360*x^3 + 2809180*x^2 - 205100*x + 1267924)
 
gp: K = bnfinit(x^20 + 55*x^18 + 1290*x^16 - 4*x^15 + 16820*x^14 + 70*x^13 + 134325*x^12 + 3810*x^11 + 681153*x^10 + 36060*x^9 + 2183600*x^8 + 96710*x^7 + 4242720*x^6 - 105998*x^5 + 4621740*x^4 - 588360*x^3 + 2809180*x^2 - 205100*x + 1267924, 1)
 

Normalized defining polynomial

\( x^{20} + 55 x^{18} + 1290 x^{16} - 4 x^{15} + 16820 x^{14} + 70 x^{13} + 134325 x^{12} + 3810 x^{11} + 681153 x^{10} + 36060 x^{9} + 2183600 x^{8} + 96710 x^{7} + 4242720 x^{6} - 105998 x^{5} + 4621740 x^{4} - 588360 x^{3} + 2809180 x^{2} - 205100 x + 1267924 \)

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:  \(957979102781406250000000000000000=2^{16}\cdot 5^{22}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.57$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{2506} a^{18} - \frac{19}{1253} a^{17} + \frac{144}{1253} a^{16} - \frac{5}{1253} a^{15} - \frac{17}{2506} a^{14} + \frac{111}{1253} a^{13} + \frac{76}{1253} a^{12} - \frac{141}{2506} a^{11} - \frac{267}{1253} a^{10} + \frac{319}{1253} a^{9} - \frac{395}{1253} a^{8} - \frac{157}{358} a^{7} - \frac{284}{1253} a^{6} - \frac{269}{1253} a^{5} - \frac{422}{1253} a^{4} + \frac{606}{1253} a^{3} - \frac{318}{1253} a^{2} + \frac{221}{1253} a - \frac{68}{179}$, $\frac{1}{3756768078576991982518799763917817859696718153723166} a^{19} - \frac{641673428473206492292237233735434582055686015489}{3756768078576991982518799763917817859696718153723166} a^{18} + \frac{104594043176269131066827031778346049511979938047533}{536681154082427426074114251988259694242388307674738} a^{17} + \frac{357033860462275405914280477293426830110154499245941}{1878384039288495991259399881958908929848359076861583} a^{16} - \frac{79650182398824646101671475887426673080269228103903}{1878384039288495991259399881958908929848359076861583} a^{15} + \frac{575500040332612380893179023655625344246262731913167}{3756768078576991982518799763917817859696718153723166} a^{14} + \frac{87098994849864104442321587939016706509072876547937}{1252256026192330660839599921305939286565572717907722} a^{13} + \frac{353632016932528818931034739970302472181969519250895}{3756768078576991982518799763917817859696718153723166} a^{12} - \frac{9388539976173841221164979841667928461446874993827}{268340577041213713037057125994129847121194153837369} a^{11} + \frac{137733780715724058967330907517635028439338927893887}{3756768078576991982518799763917817859696718153723166} a^{10} + \frac{1617412340215984059512703594939479076982036511469445}{3756768078576991982518799763917817859696718153723166} a^{9} - \frac{1819683601985223151120250497156336450552594876751867}{3756768078576991982518799763917817859696718153723166} a^{8} + \frac{166666381907509987917104303101774129101098161801763}{1252256026192330660839599921305939286565572717907722} a^{7} + \frac{848943284567401988617654662617951726014700016886835}{3756768078576991982518799763917817859696718153723166} a^{6} - \frac{1207412601215889479511837785112846808622648201326687}{3756768078576991982518799763917817859696718153723166} a^{5} + \frac{151228801593974581135899741154364814290204532009927}{626128013096165330419799960652969643282786358953861} a^{4} - \frac{144392655446293228868054290555670643586935584788090}{626128013096165330419799960652969643282786358953861} a^{3} + \frac{172172460374591811891712399364728653654408639794210}{626128013096165330419799960652969643282786358953861} a^{2} - \frac{332033993454623384357405027816579582913415601971042}{1878384039288495991259399881958908929848359076861583} a + \frac{9008037723808216799679357347975868307448951738563}{268340577041213713037057125994129847121194153837369}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 81597325.15615189 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{5}, \sqrt{-19})\), 5.1.50000.1, 10.2.12500000000.1, 10.0.6190247500000000.1, 10.0.30951237500000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$