↑

Properties

Label 20.10.4075596666...9375.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,3^{8}\cdot 5^{12}\cdot 239^{9}$
Root discriminant $47.92$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 216, -738, -891, 2181, -5613, -18394, 216, 18905, -114, -7955, 3185, 1216, -1754, 302, 221, -97, -9, 25, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 25*x^18 - 9*x^17 - 97*x^16 + 221*x^15 + 302*x^14 - 1754*x^13 + 1216*x^12 + 3185*x^11 - 7955*x^10 - 114*x^9 + 18905*x^8 + 216*x^7 - 18394*x^6 - 5613*x^5 + 2181*x^4 - 891*x^3 - 738*x^2 + 216*x + 81)
 
gp: K = bnfinit(x^20 - 9*x^19 + 25*x^18 - 9*x^17 - 97*x^16 + 221*x^15 + 302*x^14 - 1754*x^13 + 1216*x^12 + 3185*x^11 - 7955*x^10 - 114*x^9 + 18905*x^8 + 216*x^7 - 18394*x^6 - 5613*x^5 + 2181*x^4 - 891*x^3 - 738*x^2 + 216*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 25 x^{18} - 9 x^{17} - 97 x^{16} + 221 x^{15} + 302 x^{14} - 1754 x^{13} + 1216 x^{12} + 3185 x^{11} - 7955 x^{10} - 114 x^{9} + 18905 x^{8} + 216 x^{7} - 18394 x^{6} - 5613 x^{5} + 2181 x^{4} - 891 x^{3} - 738 x^{2} + 216 x + 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4075596666226495465452050537109375=-\,3^{8}\cdot 5^{12}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.92$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{15} a^{16} + \frac{1}{15} a^{14} - \frac{2}{5} a^{13} - \frac{1}{15} a^{12} + \frac{2}{15} a^{11} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{2}{15} a^{8} - \frac{7}{15} a^{7} + \frac{1}{15} a^{6} - \frac{2}{5} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{45} a^{17} + \frac{4}{45} a^{15} + \frac{8}{45} a^{13} - \frac{22}{45} a^{12} + \frac{17}{45} a^{11} + \frac{4}{45} a^{10} + \frac{22}{45} a^{9} + \frac{1}{9} a^{8} - \frac{2}{45} a^{7} + \frac{2}{15} a^{6} + \frac{11}{45} a^{5} + \frac{2}{5} a^{4} - \frac{13}{45} a^{3} + \frac{1}{15} a^{2} - \frac{1}{15} a - \frac{1}{5}$, $\frac{1}{135} a^{18} + \frac{4}{135} a^{16} - \frac{1}{15} a^{15} + \frac{8}{135} a^{14} - \frac{67}{135} a^{13} + \frac{17}{135} a^{12} - \frac{23}{135} a^{11} + \frac{8}{27} a^{10} - \frac{8}{27} a^{9} - \frac{13}{27} a^{8} - \frac{19}{45} a^{7} - \frac{52}{135} a^{6} + \frac{2}{15} a^{5} + \frac{1}{27} a^{4} - \frac{17}{45} a^{3} - \frac{19}{45} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{241221452532790667392136550825} a^{19} - \frac{24370155862480914519660887}{241221452532790667392136550825} a^{18} + \frac{28359307846267414557825491}{21929222957526424308376050075} a^{17} + \frac{882233153175323746815669433}{241221452532790667392136550825} a^{16} + \frac{17169023619992179835407803224}{241221452532790667392136550825} a^{15} - \frac{10592097148890740200364712371}{241221452532790667392136550825} a^{14} - \frac{10998110885406680068808442598}{48244290506558133478427310165} a^{13} + \frac{9080489000997614527888949407}{80407150844263555797378850275} a^{12} + \frac{40134211531759543151347704848}{241221452532790667392136550825} a^{11} + \frac{395586497175309108252752419}{26802383614754518599126283425} a^{10} + \frac{3675457634393951036659134721}{8934127871584839533042094475} a^{9} - \frac{11186175282536990795506420396}{48244290506558133478427310165} a^{8} - \frac{21242270587911921894383245463}{48244290506558133478427310165} a^{7} - \frac{99764274168003473678145206119}{241221452532790667392136550825} a^{6} + \frac{69329047221634405116312533423}{241221452532790667392136550825} a^{5} - \frac{47104727615027252034588626137}{241221452532790667392136550825} a^{4} - \frac{22621279246224729860668303786}{80407150844263555797378850275} a^{3} - \frac{37756181015737338164695600549}{80407150844263555797378850275} a^{2} - \frac{451083662186782109828429653}{26802383614754518599126283425} a + \frac{1992039608952029272917543321}{8934127871584839533042094475}$

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:  $14$
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:  \( 3190693194.86 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
239Data not computed