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Properties

Label 20.10.2034837638...9375.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{10}\cdot 27517559^{3}$
Root discriminant $29.20$
Ramified primes $5, 27517559$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![619, -2902, -1589, 14615, 18, -24322, 7904, 19019, -14841, -4943, 10512, -2441, -3133, 2165, 131, -605, 131, 65, -23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 23*x^18 + 65*x^17 + 131*x^16 - 605*x^15 + 131*x^14 + 2165*x^13 - 3133*x^12 - 2441*x^11 + 10512*x^10 - 4943*x^9 - 14841*x^8 + 19019*x^7 + 7904*x^6 - 24322*x^5 + 18*x^4 + 14615*x^3 - 1589*x^2 - 2902*x + 619)
 
gp: K = bnfinit(x^20 - 2*x^19 - 23*x^18 + 65*x^17 + 131*x^16 - 605*x^15 + 131*x^14 + 2165*x^13 - 3133*x^12 - 2441*x^11 + 10512*x^10 - 4943*x^9 - 14841*x^8 + 19019*x^7 + 7904*x^6 - 24322*x^5 + 18*x^4 + 14615*x^3 - 1589*x^2 - 2902*x + 619, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 23 x^{18} + 65 x^{17} + 131 x^{16} - 605 x^{15} + 131 x^{14} + 2165 x^{13} - 3133 x^{12} - 2441 x^{11} + 10512 x^{10} - 4943 x^{9} - 14841 x^{8} + 19019 x^{7} + 7904 x^{6} - 24322 x^{5} + 18 x^{4} + 14615 x^{3} - 1589 x^{2} - 2902 x + 619 \)

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:  \(-203483763895883268631630859375=-\,5^{10}\cdot 27517559^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.20$
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:  $5, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{12907062377531321595780375338628701519} a^{19} + \frac{1112536619106679935845190140025884893}{12907062377531321595780375338628701519} a^{18} + \frac{3040368226226533217796661367707451087}{12907062377531321595780375338628701519} a^{17} - \frac{111753063204013613063614585529963782}{12907062377531321595780375338628701519} a^{16} - \frac{4570439087816451778182530799759531655}{12907062377531321595780375338628701519} a^{15} - \frac{4118145308629530929315186237030926698}{12907062377531321595780375338628701519} a^{14} + \frac{5217135507212956977931361856619706959}{12907062377531321595780375338628701519} a^{13} - \frac{120202011157084337748005762319638963}{12907062377531321595780375338628701519} a^{12} - \frac{4061168644894388199011593878974475137}{12907062377531321595780375338628701519} a^{11} - \frac{5920339180731726529704462981026998850}{12907062377531321595780375338628701519} a^{10} + \frac{5407619211611530725490487704176744000}{12907062377531321595780375338628701519} a^{9} - \frac{1134415866959623036479085867741051920}{12907062377531321595780375338628701519} a^{8} - \frac{2510083899536974532812457829034431671}{12907062377531321595780375338628701519} a^{7} - \frac{570790327696638807954833027199253883}{12907062377531321595780375338628701519} a^{6} + \frac{5582149079243243120761837974824470309}{12907062377531321595780375338628701519} a^{5} + \frac{3590144086824733963930439965337067052}{12907062377531321595780375338628701519} a^{4} - \frac{4220045043847358693569646767205061976}{12907062377531321595780375338628701519} a^{3} - \frac{3742700722712173802207699907442453906}{12907062377531321595780375338628701519} a^{2} + \frac{377593633238599702865679952886287830}{992850952117793968906182718356053963} a + \frac{2027197621071037365396871385529627510}{12907062377531321595780375338628701519}$

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:  \( 13989511.0388 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.85992371875.1

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 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
27517559Data not computed