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Properties

Label 20.0.42390388623...8576.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 401^{8}$
Root discriminant $53.87$
Ramified primes $2, 3, 401$
Class number $192$ (GRH)
Class group $[2, 2, 48]$ (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![244053, -615708, 964152, -1623348, 2265462, -2490132, 2518702, -2307508, 1791009, -1218710, 740054, -389440, 180306, -74414, 26400, -8164, 2186, -468, 86, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 86*x^18 - 468*x^17 + 2186*x^16 - 8164*x^15 + 26400*x^14 - 74414*x^13 + 180306*x^12 - 389440*x^11 + 740054*x^10 - 1218710*x^9 + 1791009*x^8 - 2307508*x^7 + 2518702*x^6 - 2490132*x^5 + 2265462*x^4 - 1623348*x^3 + 964152*x^2 - 615708*x + 244053)
 
gp: K = bnfinit(x^20 - 10*x^19 + 86*x^18 - 468*x^17 + 2186*x^16 - 8164*x^15 + 26400*x^14 - 74414*x^13 + 180306*x^12 - 389440*x^11 + 740054*x^10 - 1218710*x^9 + 1791009*x^8 - 2307508*x^7 + 2518702*x^6 - 2490132*x^5 + 2265462*x^4 - 1623348*x^3 + 964152*x^2 - 615708*x + 244053, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 86 x^{18} - 468 x^{17} + 2186 x^{16} - 8164 x^{15} + 26400 x^{14} - 74414 x^{13} + 180306 x^{12} - 389440 x^{11} + 740054 x^{10} - 1218710 x^{9} + 1791009 x^{8} - 2307508 x^{7} + 2518702 x^{6} - 2490132 x^{5} + 2265462 x^{4} - 1623348 x^{3} + 964152 x^{2} - 615708 x + 244053 \)

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:  \(42390388623295510152536086938648576=2^{30}\cdot 3^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.87$
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:  $2, 3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{4}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{207} a^{17} - \frac{5}{207} a^{16} - \frac{5}{69} a^{15} + \frac{20}{207} a^{14} + \frac{3}{23} a^{13} - \frac{4}{69} a^{12} + \frac{8}{23} a^{11} - \frac{98}{207} a^{10} + \frac{62}{207} a^{9} - \frac{2}{9} a^{8} - \frac{77}{207} a^{7} + \frac{73}{207} a^{6} - \frac{17}{207} a^{5} + \frac{2}{23} a^{3} + \frac{8}{69} a^{2} - \frac{10}{23} a$, $\frac{1}{979317} a^{18} - \frac{697}{326439} a^{17} + \frac{32288}{979317} a^{16} + \frac{10058}{979317} a^{15} + \frac{162271}{979317} a^{14} + \frac{5464}{326439} a^{13} - \frac{32480}{326439} a^{12} - \frac{343214}{979317} a^{11} - \frac{485579}{979317} a^{10} - \frac{151318}{326439} a^{9} - \frac{58888}{979317} a^{8} - \frac{162221}{326439} a^{7} + \frac{130567}{326439} a^{6} - \frac{165742}{979317} a^{5} - \frac{14129}{36271} a^{4} - \frac{35848}{108813} a^{3} + \frac{1450}{108813} a^{2} - \frac{48922}{108813} a - \frac{588}{1577}$, $\frac{1}{28743528411520503437162691633214313703} a^{19} + \frac{9334638061043434848452791906078}{28743528411520503437162691633214313703} a^{18} - \frac{19980879413711325383129664904561867}{28743528411520503437162691633214313703} a^{17} + \frac{43248017839988991479167773043386350}{1249718626587847975528812679704970161} a^{16} - \frac{18058958290545188676134376479805245}{187866198768107865602370533550420351} a^{15} - \frac{808725638801072557708862784009384674}{28743528411520503437162691633214313703} a^{14} + \frac{617601920747095034604216537353964437}{9581176137173501145720897211071437901} a^{13} - \frac{4177943396317592676150647481048034166}{28743528411520503437162691633214313703} a^{12} - \frac{2768910706777318753279454320104464296}{28743528411520503437162691633214313703} a^{11} + \frac{544874382290949820257308353568077270}{1512817284816868601955931138590227037} a^{10} - \frac{7715756697646905360016524635432714791}{28743528411520503437162691633214313703} a^{9} + \frac{8751675381363080573360325102147253592}{28743528411520503437162691633214313703} a^{8} - \frac{839786939534250875441763204949894732}{3193725379057833715240299070357145967} a^{7} - \frac{737150313052345032815412607062871942}{1512817284816868601955931138590227037} a^{6} + \frac{4433738970948139078087421863013184359}{28743528411520503437162691633214313703} a^{5} + \frac{1329717927225982931688862727733019981}{3193725379057833715240299070357145967} a^{4} - \frac{1307003648049611184951501704632838578}{3193725379057833715240299070357145967} a^{3} - \frac{49670273799855417259735307825385347}{3193725379057833715240299070357145967} a^{2} + \frac{1326338510831521237662514078132859300}{3193725379057833715240299070357145967} a - \frac{15238325163029033725114417814177166}{46285875058809184278844914063147043}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{48}$, which has order $192$ (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:  \( 15405657.6163 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 5.5.160801.1, 10.0.2144679823033344.1, 10.10.6434039469100032.1, 10.0.79432586038272.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{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}$
401Data not computed