↑

Properties

Label 20.12.2024969848...1853.1
Degree $20$
Signature $[12, 4]$
Discriminant $13^{13}\cdot 401^{8}$
Root discriminant $58.25$
Ramified primes $13, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-187353, 771768, 94932, -1406673, 631395, 819072, -769264, -268437, 238607, 115845, -7584, -31062, -9141, 1518, 1905, 501, -102, -48, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 - 48*x^17 - 102*x^16 + 501*x^15 + 1905*x^14 + 1518*x^13 - 9141*x^12 - 31062*x^11 - 7584*x^10 + 115845*x^9 + 238607*x^8 - 268437*x^7 - 769264*x^6 + 819072*x^5 + 631395*x^4 - 1406673*x^3 + 94932*x^2 + 771768*x - 187353)
 
gp: K = bnfinit(x^20 - 11*x^18 - 48*x^17 - 102*x^16 + 501*x^15 + 1905*x^14 + 1518*x^13 - 9141*x^12 - 31062*x^11 - 7584*x^10 + 115845*x^9 + 238607*x^8 - 268437*x^7 - 769264*x^6 + 819072*x^5 + 631395*x^4 - 1406673*x^3 + 94932*x^2 + 771768*x - 187353, 1)
 

Normalized defining polynomial

\( x^{20} - 11 x^{18} - 48 x^{17} - 102 x^{16} + 501 x^{15} + 1905 x^{14} + 1518 x^{13} - 9141 x^{12} - 31062 x^{11} - 7584 x^{10} + 115845 x^{9} + 238607 x^{8} - 268437 x^{7} - 769264 x^{6} + 819072 x^{5} + 631395 x^{4} - 1406673 x^{3} + 94932 x^{2} + 771768 x - 187353 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(202496984818189934420549080247241853=13^{13}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.25$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{6} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{7} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{8} - \frac{2}{9} a^{4}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{9} + \frac{1}{9} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{2}$, $\frac{1}{27} a^{15} - \frac{1}{9} a^{7} + \frac{2}{27} a^{3}$, $\frac{1}{27} a^{16} - \frac{1}{9} a^{8} + \frac{2}{27} a^{4}$, $\frac{1}{27} a^{17} - \frac{1}{9} a^{9} + \frac{2}{27} a^{5}$, $\frac{1}{81} a^{18} + \frac{1}{81} a^{16} - \frac{1}{27} a^{14} - \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{27} a^{8} - \frac{1}{27} a^{7} - \frac{1}{81} a^{6} + \frac{2}{27} a^{5} - \frac{28}{81} a^{4} + \frac{11}{27} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{68605759323370132755145038239392117346263821} a^{19} + \frac{92691311924737993116563818640030353423669}{22868586441123377585048346079797372448754607} a^{18} + \frac{51081162825496953217331128484602260734650}{3610829438072112250270791486283795649803359} a^{17} + \frac{1974844580986432904465130954352401829799}{7622862147041125861682782026599124149584869} a^{16} + \frac{358464001572330782230050540010115052219356}{22868586441123377585048346079797372448754607} a^{15} + \frac{1010390737599671749991017894453994334954164}{22868586441123377585048346079797372448754607} a^{14} + \frac{734311659573222682169527729089205330120268}{22868586441123377585048346079797372448754607} a^{13} + \frac{775670940845752893468570788452367035472837}{22868586441123377585048346079797372448754607} a^{12} - \frac{939919196414711032575897499632339018815456}{22868586441123377585048346079797372448754607} a^{11} + \frac{1114844075364660459159559736718229315387814}{22868586441123377585048346079797372448754607} a^{10} - \frac{2488959741319017530820782910429061629451406}{22868586441123377585048346079797372448754607} a^{9} - \frac{2998227322351225279669513321215988925762908}{22868586441123377585048346079797372448754607} a^{8} + \frac{9454217796957531630670863575434690750397477}{68605759323370132755145038239392117346263821} a^{7} - \frac{739180325303933811135383806608328887492431}{22868586441123377585048346079797372448754607} a^{6} + \frac{1176206903198104279642784983041596905876004}{68605759323370132755145038239392117346263821} a^{5} + \frac{513249534585326046164885708923144119901784}{1203609812690704083423597162094598549934453} a^{4} - \frac{1358275065844180261436503522341445909428496}{7622862147041125861682782026599124149584869} a^{3} - \frac{1212202779170965785112704817883703201300593}{2540954049013708620560927342199708049861623} a^{2} - \frac{126740336939773032180509814306096261348620}{282328227668189846728991926911078672206847} a - \frac{26057270921545824115572654441992722398415}{282328227668189846728991926911078672206847}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 143200429240 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.160801.1, 10.10.9600508843720093.1

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

Sibling fields

Degree 20 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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
401Data not computed