↑

Properties

Label 20.0.41396863889...1024.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 401^{8}$
Root discriminant $38.09$
Ramified primes $2, 3, 401$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T73)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 0, -2484, 0, 5521, 0, -6950, 0, 6286, 0, -3848, 0, 1740, 0, -523, 0, 112, 0, -13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 13*x^18 + 112*x^16 - 523*x^14 + 1740*x^12 - 3848*x^10 + 6286*x^8 - 6950*x^6 + 5521*x^4 - 2484*x^2 + 729)
 
gp: K = bnfinit(x^20 - 13*x^18 + 112*x^16 - 523*x^14 + 1740*x^12 - 3848*x^10 + 6286*x^8 - 6950*x^6 + 5521*x^4 - 2484*x^2 + 729, 1)
 

Normalized defining polynomial

\( x^{20} - 13 x^{18} + 112 x^{16} - 523 x^{14} + 1740 x^{12} - 3848 x^{10} + 6286 x^{8} - 6950 x^{6} + 5521 x^{4} - 2484 x^{2} + 729 \)

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:  \(41396863889937021633336022401024=2^{20}\cdot 3^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.09$
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:  $2, 3, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{53728709071329} a^{18} - \frac{24800092011139}{53728709071329} a^{16} + \frac{1092255631105}{53728709071329} a^{14} + \frac{7647205234139}{53728709071329} a^{12} - \frac{4480306157183}{17909569690443} a^{10} - \frac{16541773137329}{53728709071329} a^{8} + \frac{10885905272812}{53728709071329} a^{6} + \frac{25510450001002}{53728709071329} a^{4} - \frac{15090432721673}{53728709071329} a^{2} - \frac{861605873117}{1989952187827}$, $\frac{1}{483558381641961} a^{19} + \frac{28928617060190}{483558381641961} a^{17} - \frac{160093871582882}{483558381641961} a^{15} + \frac{115104623376797}{483558381641961} a^{13} - \frac{58209015228512}{161186127213987} a^{11} + \frac{144644354076658}{483558381641961} a^{9} - \frac{150300221941175}{483558381641961} a^{7} - \frac{28218259070327}{483558381641961} a^{5} - \frac{230005269006989}{483558381641961} a^{3} - \frac{4841510248771}{17909569690443} a$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( \frac{55507540556}{53728709071329} a^{18} - \frac{688292597699}{53728709071329} a^{16} + \frac{5819991894872}{53728709071329} a^{14} - \frac{25724137012940}{53728709071329} a^{12} + \frac{27565346511287}{17909569690443} a^{10} - \frac{170293247358112}{53728709071329} a^{8} + \frac{267012110091680}{53728709071329} a^{6} - \frac{263910488265922}{53728709071329} a^{4} + \frac{221655261621788}{53728709071329} a^{2} - \frac{1651232106549}{1989952187827} \) (order $6$)
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:  \( 12890899.4691 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T73):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.160801.1, 10.0.6283241669043.1, 10.0.79432586038272.1, 10.10.2144679823033344.1

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 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed