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Properties

Label 20.0.12139714372...9424.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 13^{14}\cdot 19^{6}$
Root discriminant $25.36$
Ramified primes $2, 13, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -73, 304, -893, 1982, -3283, 4190, -4237, 3067, -968, -402, -24, 697, -329, 40, -117, 96, -31, 12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 12*x^18 - 31*x^17 + 96*x^16 - 117*x^15 + 40*x^14 - 329*x^13 + 697*x^12 - 24*x^11 - 402*x^10 - 968*x^9 + 3067*x^8 - 4237*x^7 + 4190*x^6 - 3283*x^5 + 1982*x^4 - 893*x^3 + 304*x^2 - 73*x + 9)
 
gp: K = bnfinit(x^20 - 5*x^19 + 12*x^18 - 31*x^17 + 96*x^16 - 117*x^15 + 40*x^14 - 329*x^13 + 697*x^12 - 24*x^11 - 402*x^10 - 968*x^9 + 3067*x^8 - 4237*x^7 + 4190*x^6 - 3283*x^5 + 1982*x^4 - 893*x^3 + 304*x^2 - 73*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 12 x^{18} - 31 x^{17} + 96 x^{16} - 117 x^{15} + 40 x^{14} - 329 x^{13} + 697 x^{12} - 24 x^{11} - 402 x^{10} - 968 x^{9} + 3067 x^{8} - 4237 x^{7} + 4190 x^{6} - 3283 x^{5} + 1982 x^{4} - 893 x^{3} + 304 x^{2} - 73 x + 9 \)

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:  \(12139714372817312289823719424=2^{16}\cdot 13^{14}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.36$
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:  $2, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{19581578975266004602091329638} a^{19} + \frac{4706751173565863492209099487}{19581578975266004602091329638} a^{18} - \frac{2101175578742662596152233163}{9790789487633002301045664819} a^{17} - \frac{2269947019636007346276672178}{9790789487633002301045664819} a^{16} + \frac{4278597580247557485196564309}{19581578975266004602091329638} a^{15} - \frac{3071037992204681985029453983}{9790789487633002301045664819} a^{14} + \frac{4857748666779863019209467847}{19581578975266004602091329638} a^{13} + \frac{137671895575341548752876958}{3263596495877667433681888273} a^{12} - \frac{164791397413487248306113148}{9790789487633002301045664819} a^{11} - \frac{3669925268904813290334080791}{9790789487633002301045664819} a^{10} - \frac{191712793549144348406028116}{890071771603000209185969529} a^{9} + \frac{222279614961420342445275419}{890071771603000209185969529} a^{8} + \frac{849768050696597870657238785}{6527192991755334867363776546} a^{7} + \frac{3834863378542256963822376689}{19581578975266004602091329638} a^{6} - \frac{3370137687796464213366511591}{9790789487633002301045664819} a^{5} + \frac{486095734819374774563775705}{3263596495877667433681888273} a^{4} + \frac{8133705524057063312632586951}{19581578975266004602091329638} a^{3} + \frac{942886453166476162964098297}{3263596495877667433681888273} a^{2} + \frac{1951320922143523262074197901}{19581578975266004602091329638} a - \frac{1164782333549771814073515637}{3263596495877667433681888273}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1313584.82555 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

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 t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.51376.1, 10.0.8475413230336.1, 10.0.651954863872.1, 10.6.5798966947072.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.12.10.2$x^{12} + 39 x^{6} + 676$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$