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Properties

Label 20.0.19051472146...9744.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 36497^{4}$
Root discriminant $23.12$
Ramified primes $2, 36497$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T279

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -22, 51, -200, 524, -716, 683, -764, 1318, -1702, 1443, -834, 492, -384, 286, -148, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 148*x^17 + 286*x^16 - 384*x^15 + 492*x^14 - 834*x^13 + 1443*x^12 - 1702*x^11 + 1318*x^10 - 764*x^9 + 683*x^8 - 716*x^7 + 524*x^6 - 200*x^5 + 51*x^4 - 22*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 148*x^17 + 286*x^16 - 384*x^15 + 492*x^14 - 834*x^13 + 1443*x^12 - 1702*x^11 + 1318*x^10 - 764*x^9 + 683*x^8 - 716*x^7 + 524*x^6 - 200*x^5 + 51*x^4 - 22*x^3 + 18*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 148 x^{17} + 286 x^{16} - 384 x^{15} + 492 x^{14} - 834 x^{13} + 1443 x^{12} - 1702 x^{11} + 1318 x^{10} - 764 x^{9} + 683 x^{8} - 716 x^{7} + 524 x^{6} - 200 x^{5} + 51 x^{4} - 22 x^{3} + 18 x^{2} - 6 x + 1 \)

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:  \(1905147214615868315998879744=2^{30}\cdot 36497^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.12$
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:  $2, 36497$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{114463443183303143601} a^{19} + \frac{2734188190303420429}{12718160353700349289} a^{18} + \frac{21828080506013432915}{114463443183303143601} a^{17} + \frac{36206839419828947164}{114463443183303143601} a^{16} - \frac{4486530849951514303}{114463443183303143601} a^{15} - \frac{27017180764031251924}{114463443183303143601} a^{14} + \frac{17638277062678993034}{114463443183303143601} a^{13} + \frac{4515145946321980634}{16351920454757591943} a^{12} - \frac{2610770565421912585}{114463443183303143601} a^{11} - \frac{12186858477466175288}{114463443183303143601} a^{10} - \frac{14088795702139115002}{114463443183303143601} a^{9} - \frac{2048816288977001767}{12718160353700349289} a^{8} - \frac{27649310718000642949}{114463443183303143601} a^{7} + \frac{543813989220054337}{38154481061101047867} a^{6} + \frac{28824054968984350064}{114463443183303143601} a^{5} + \frac{826719369087152449}{5450640151585863981} a^{4} + \frac{1019483486744676745}{12718160353700349289} a^{3} - \frac{46130480344392649096}{114463443183303143601} a^{2} + \frac{2192180047415925833}{16351920454757591943} a + \frac{7471363563032634413}{114463443183303143601}$

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:  \( \frac{48769350165152192759}{114463443183303143601} a^{19} - \frac{53016340414589783940}{12718160353700349289} a^{18} + \frac{2336096768030825669476}{114463443183303143601} a^{17} - \frac{6719071075727571993550}{114463443183303143601} a^{16} + \frac{12518825248921389213226}{114463443183303143601} a^{15} - \frac{16058611410267439497887}{114463443183303143601} a^{14} + \frac{20520664012837773617125}{114463443183303143601} a^{13} - \frac{5159466533433375040559}{16351920454757591943} a^{12} + \frac{62414470916379720468394}{114463443183303143601} a^{11} - \frac{69481235016185951788972}{114463443183303143601} a^{10} + \frac{49179959767792798651354}{114463443183303143601} a^{9} - \frac{2872888862951043299306}{12718160353700349289} a^{8} + \frac{26416990876292992066135}{114463443183303143601} a^{7} - \frac{9384655571081098358455}{38154481061101047867} a^{6} + \frac{19172793700848781502071}{114463443183303143601} a^{5} - \frac{251376733735181185153}{5450640151585863981} a^{4} + \frac{85552017507786813838}{12718160353700349289} a^{3} - \frac{284955264022828097828}{114463443183303143601} a^{2} + \frac{95060144749867430266}{16351920454757591943} a - \frac{140042719842816329636}{114463443183303143601} \) (order $4$)
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:  \( 104792.356058 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T279:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.36497.1, 10.0.1363999753216.1, 10.10.1363999753216.1, 10.0.1363999753216.2

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
36497Data not computed