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Properties

Label 20.12.5834302965...2368.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{10}\cdot 19^{10}\cdot 43^{11}$
Root discriminant $48.79$
Ramified primes $2, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -2500, -4300, 10775, 8721, 43458, -60667, 10703, 10908, -14075, 18559, -15674, 4324, 4404, -4324, 1401, -147, -28, 23, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 23*x^18 - 28*x^17 - 147*x^16 + 1401*x^15 - 4324*x^14 + 4404*x^13 + 4324*x^12 - 15674*x^11 + 18559*x^10 - 14075*x^9 + 10908*x^8 + 10703*x^7 - 60667*x^6 + 43458*x^5 + 8721*x^4 + 10775*x^3 - 4300*x^2 - 2500*x + 625)
 
gp: K = bnfinit(x^20 - 8*x^19 + 23*x^18 - 28*x^17 - 147*x^16 + 1401*x^15 - 4324*x^14 + 4404*x^13 + 4324*x^12 - 15674*x^11 + 18559*x^10 - 14075*x^9 + 10908*x^8 + 10703*x^7 - 60667*x^6 + 43458*x^5 + 8721*x^4 + 10775*x^3 - 4300*x^2 - 2500*x + 625, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 23 x^{18} - 28 x^{17} - 147 x^{16} + 1401 x^{15} - 4324 x^{14} + 4404 x^{13} + 4324 x^{12} - 15674 x^{11} + 18559 x^{10} - 14075 x^{9} + 10908 x^{8} + 10703 x^{7} - 60667 x^{6} + 43458 x^{5} + 8721 x^{4} + 10775 x^{3} - 4300 x^{2} - 2500 x + 625 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.79$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{65} a^{16} - \frac{6}{65} a^{15} + \frac{21}{65} a^{14} + \frac{14}{65} a^{13} + \frac{2}{5} a^{12} - \frac{22}{65} a^{11} + \frac{2}{65} a^{10} - \frac{2}{65} a^{9} + \frac{6}{13} a^{8} + \frac{31}{65} a^{7} + \frac{21}{65} a^{6} + \frac{32}{65} a^{5} - \frac{1}{5} a^{4} + \frac{17}{65} a^{3} + \frac{2}{65} a^{2} + \frac{32}{65} a + \frac{2}{13}$, $\frac{1}{65} a^{17} - \frac{3}{13} a^{15} + \frac{2}{13} a^{14} - \frac{4}{13} a^{13} + \frac{4}{65} a^{12} + \frac{2}{13} a^{10} + \frac{18}{65} a^{9} + \frac{16}{65} a^{8} + \frac{12}{65} a^{7} + \frac{28}{65} a^{6} - \frac{16}{65} a^{5} + \frac{4}{65} a^{4} - \frac{2}{5} a^{3} - \frac{21}{65} a^{2} + \frac{7}{65} a - \frac{1}{13}$, $\frac{1}{1625} a^{18} - \frac{8}{1625} a^{17} - \frac{2}{1625} a^{16} + \frac{44}{125} a^{15} + \frac{628}{1625} a^{14} - \frac{499}{1625} a^{13} - \frac{799}{1625} a^{12} - \frac{471}{1625} a^{11} - \frac{751}{1625} a^{10} - \frac{349}{1625} a^{9} + \frac{534}{1625} a^{8} + \frac{16}{65} a^{7} - \frac{292}{1625} a^{6} + \frac{353}{1625} a^{5} + \frac{683}{1625} a^{4} - \frac{567}{1625} a^{3} + \frac{71}{1625} a^{2} - \frac{6}{13} a - \frac{1}{65}$, $\frac{1}{37598313255771286400091256813994918125} a^{19} + \frac{8900638902718255272444577736145277}{37598313255771286400091256813994918125} a^{18} - \frac{133115678314323275702426666822637982}{37598313255771286400091256813994918125} a^{17} + \frac{211212416086683215022974238421612702}{37598313255771286400091256813994918125} a^{16} - \frac{14655479674198061474059383384452270677}{37598313255771286400091256813994918125} a^{15} + \frac{3149054843076551213733787789221255056}{37598313255771286400091256813994918125} a^{14} + \frac{15881207436941144262380764888948260661}{37598313255771286400091256813994918125} a^{13} + \frac{16039935289462367857337462282299530964}{37598313255771286400091256813994918125} a^{12} + \frac{18374707384695072581529982417907096239}{37598313255771286400091256813994918125} a^{11} - \frac{13789342024042067463417036297336368859}{37598313255771286400091256813994918125} a^{10} - \frac{10887312389379722782765240873233739556}{37598313255771286400091256813994918125} a^{9} - \frac{286681257046649321036089525240696694}{578435588550327483078327027907614125} a^{8} + \frac{3207696044190250132535752454116090883}{37598313255771286400091256813994918125} a^{7} + \frac{9635729686338801970958914456916751858}{37598313255771286400091256813994918125} a^{6} - \frac{5507760440947875182432876535923904612}{37598313255771286400091256813994918125} a^{5} - \frac{13001758079723633734876476496634670937}{37598313255771286400091256813994918125} a^{4} - \frac{4234419697202658980596862366539387424}{37598313255771286400091256813994918125} a^{3} - \frac{1230279003654288412498081745512484458}{7519662651154257280018251362798983625} a^{2} + \frac{113247041797155988435777795323909319}{1503932530230851456003650272559796725} a - \frac{120491026307764821716452567360712462}{300786506046170291200730054511959345}$

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:  $15$
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:  \( 5540823478.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.9$x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$