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Properties

Label 20.0.82270188117...0144.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 11^{17}\cdot 23^{6}$
Root discriminant $78.66$
Ramified primes $2, 11, 23$
Class number $15216$ (GRH)
Class group $[2, 2, 3804]$ (GRH)
Galois group 20T326

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![69135539, 0, 168727724, 0, 169077370, 0, 92413816, 0, 30638916, 0, 6447376, 0, 870719, 0, 74104, 0, 3758, 0, 100, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 100*x^18 + 3758*x^16 + 74104*x^14 + 870719*x^12 + 6447376*x^10 + 30638916*x^8 + 92413816*x^6 + 169077370*x^4 + 168727724*x^2 + 69135539)
 
gp: K = bnfinit(x^20 + 100*x^18 + 3758*x^16 + 74104*x^14 + 870719*x^12 + 6447376*x^10 + 30638916*x^8 + 92413816*x^6 + 169077370*x^4 + 168727724*x^2 + 69135539, 1)
 

Normalized defining polynomial

\( x^{20} + 100 x^{18} + 3758 x^{16} + 74104 x^{14} + 870719 x^{12} + 6447376 x^{10} + 30638916 x^{8} + 92413816 x^{6} + 169077370 x^{4} + 168727724 x^{2} + 69135539 \)

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:  \(82270188117030423168911138645967110144=2^{40}\cdot 11^{17}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.66$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} - \frac{4}{23} a^{12} + \frac{11}{23} a^{10} + \frac{4}{23} a^{8} + \frac{6}{23} a^{6} - \frac{10}{23} a^{4}$, $\frac{1}{23} a^{15} - \frac{4}{23} a^{13} + \frac{11}{23} a^{11} + \frac{4}{23} a^{9} + \frac{6}{23} a^{7} - \frac{10}{23} a^{5}$, $\frac{1}{23} a^{16} - \frac{5}{23} a^{12} + \frac{2}{23} a^{10} - \frac{1}{23} a^{8} - \frac{9}{23} a^{6} + \frac{6}{23} a^{4}$, $\frac{1}{23} a^{17} - \frac{5}{23} a^{13} + \frac{2}{23} a^{11} - \frac{1}{23} a^{9} - \frac{9}{23} a^{7} + \frac{6}{23} a^{5}$, $\frac{1}{573141965179360297211921} a^{18} - \frac{4201016532314612270073}{573141965179360297211921} a^{16} - \frac{7221003464091661283189}{573141965179360297211921} a^{14} - \frac{109833612925393308807458}{573141965179360297211921} a^{12} + \frac{41511128376777762137100}{573141965179360297211921} a^{10} - \frac{275244939203799063246487}{573141965179360297211921} a^{8} + \frac{37806841441690205216725}{573141965179360297211921} a^{6} - \frac{260736171765237952642247}{573141965179360297211921} a^{4} - \frac{9927668681932185168656}{24919215877363491183127} a^{2} - \frac{7995372277696634873634}{24919215877363491183127}$, $\frac{1}{62472474204550272396099389} a^{19} + \frac{593860164524409176124975}{62472474204550272396099389} a^{17} + \frac{765274688734176565393748}{62472474204550272396099389} a^{15} - \frac{7336406217360805751914288}{62472474204550272396099389} a^{13} - \frac{27519141631987243486401362}{62472474204550272396099389} a^{11} + \frac{20557219534272079565847685}{62472474204550272396099389} a^{9} - \frac{13892034834004501366151268}{62472474204550272396099389} a^{7} + \frac{4772945435462187266349407}{62472474204550272396099389} a^{5} - \frac{308958259210294079366180}{2716194530632620538960843} a^{3} + \frac{515308161146936679972033}{2716194530632620538960843} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3804}$, which has order $15216$ (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:  \( 2545371.69018 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T326:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n326 are not computed
Character table for t20n326 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.2

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$