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Properties

Label 20.20.1368902819...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{40}\cdot 5^{20}\cdot 97^{2}\cdot 193^{4}$
Root discriminant $90.54$
Ramified primes $2, 5, 97, 193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34, 4900, -20390, -70720, 399385, -197964, -970350, 672640, 1085935, -627620, -680984, 237960, 232690, -33060, -37030, 1840, 2760, -40, -90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 90*x^18 - 40*x^17 + 2760*x^16 + 1840*x^15 - 37030*x^14 - 33060*x^13 + 232690*x^12 + 237960*x^11 - 680984*x^10 - 627620*x^9 + 1085935*x^8 + 672640*x^7 - 970350*x^6 - 197964*x^5 + 399385*x^4 - 70720*x^3 - 20390*x^2 + 4900*x + 34)
 
gp: K = bnfinit(x^20 - 90*x^18 - 40*x^17 + 2760*x^16 + 1840*x^15 - 37030*x^14 - 33060*x^13 + 232690*x^12 + 237960*x^11 - 680984*x^10 - 627620*x^9 + 1085935*x^8 + 672640*x^7 - 970350*x^6 - 197964*x^5 + 399385*x^4 - 70720*x^3 - 20390*x^2 + 4900*x + 34, 1)
 

Normalized defining polynomial

\( x^{20} - 90 x^{18} - 40 x^{17} + 2760 x^{16} + 1840 x^{15} - 37030 x^{14} - 33060 x^{13} + 232690 x^{12} + 237960 x^{11} - 680984 x^{10} - 627620 x^{9} + 1085935 x^{8} + 672640 x^{7} - 970350 x^{6} - 197964 x^{5} + 399385 x^{4} - 70720 x^{3} - 20390 x^{2} + 4900 x + 34 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1368902819004704358400000000000000000000=2^{40}\cdot 5^{20}\cdot 97^{2}\cdot 193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.54$
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:  $2, 5, 97, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{10} + \frac{2}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{11} + \frac{2}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{12} + \frac{2}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{3062822995} a^{18} + \frac{216120909}{3062822995} a^{17} + \frac{166634872}{3062822995} a^{16} - \frac{221405138}{3062822995} a^{15} + \frac{181111197}{612564599} a^{14} + \frac{877913994}{3062822995} a^{13} + \frac{938900741}{3062822995} a^{12} - \frac{665983497}{3062822995} a^{11} - \frac{461335472}{3062822995} a^{10} + \frac{196989461}{612564599} a^{9} + \frac{1526630657}{3062822995} a^{8} + \frac{361898548}{3062822995} a^{7} - \frac{1385972931}{3062822995} a^{6} + \frac{1032106699}{3062822995} a^{5} + \frac{229797764}{612564599} a^{4} + \frac{865403459}{3062822995} a^{3} + \frac{311083201}{3062822995} a^{2} - \frac{296352187}{3062822995} a + \frac{789602468}{3062822995}$, $\frac{1}{196703260469107247527204015470846985} a^{19} - \frac{16742590986210672926356939}{196703260469107247527204015470846985} a^{18} + \frac{222478432600112919902760265556818}{4574494429514122035516372452810395} a^{17} + \frac{18890881579389682961355384660869489}{196703260469107247527204015470846985} a^{16} + \frac{7900875532419249333424308244820312}{196703260469107247527204015470846985} a^{15} + \frac{88854784083981754734138328636646814}{196703260469107247527204015470846985} a^{14} + \frac{76748669463135609398842618108365839}{196703260469107247527204015470846985} a^{13} + \frac{5944386562166405214109908265856473}{28100465781301035361029145067263855} a^{12} - \frac{89569932953027947021412895610814684}{196703260469107247527204015470846985} a^{11} + \frac{90631827137262418671774646636884003}{196703260469107247527204015470846985} a^{10} + \frac{34918849397566293216150711516981437}{196703260469107247527204015470846985} a^{9} - \frac{98131678662777134131575454850292753}{196703260469107247527204015470846985} a^{8} + \frac{85688604324661899352466596748404658}{196703260469107247527204015470846985} a^{7} + \frac{29327441898458507954616552536656448}{196703260469107247527204015470846985} a^{6} + \frac{48809840189513168084645082536648284}{196703260469107247527204015470846985} a^{5} - \frac{45353049039250757179402604733463401}{196703260469107247527204015470846985} a^{4} - \frac{35048354490576114643963438078669561}{196703260469107247527204015470846985} a^{3} - \frac{24943277655250411418393604860635924}{196703260469107247527204015470846985} a^{2} + \frac{74394093362922742061277755569918166}{196703260469107247527204015470846985} a - \frac{67102796449912484963816017695517457}{196703260469107247527204015470846985}$

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:  $19$
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:  \( 33017072432200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.11919680000000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.10.9$x^{10} + 10 x^{8} + 10 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 17$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
5.10.10.9$x^{10} + 10 x^{8} + 10 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 17$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
97Data not computed
$193$$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.4.2.2$x^{4} - 193 x^{2} + 186245$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
193.4.2.2$x^{4} - 193 x^{2} + 186245$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
193.8.0.1$x^{8} - x + 5$$1$$8$$0$$C_8$$[\ ]^{8}$