↑

Properties

Label 20.0.38698352640...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 3^{10}\cdot 5^{22}$
Root discriminant $37.97$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\times F_5$ (as 20T13)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![295975, -369350, 1027925, -1691250, 1836975, -1347420, 714510, -232740, 65655, -5630, 841, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 + 841*x^10 - 5630*x^9 + 65655*x^8 - 232740*x^7 + 714510*x^6 - 1347420*x^5 + 1836975*x^4 - 1691250*x^3 + 1027925*x^2 - 369350*x + 295975)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 + 841*x^10 - 5630*x^9 + 65655*x^8 - 232740*x^7 + 714510*x^6 - 1347420*x^5 + 1836975*x^4 - 1691250*x^3 + 1027925*x^2 - 369350*x + 295975, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 105 x^{16} + 228 x^{15} + 90 x^{14} - 540 x^{13} + 45 x^{12} + 770 x^{11} + 841 x^{10} - 5630 x^{9} + 65655 x^{8} - 232740 x^{7} + 714510 x^{6} - 1347420 x^{5} + 1836975 x^{4} - 1691250 x^{3} + 1027925 x^{2} - 369350 x + 295975 \)

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:  \(38698352640000000000000000000000=2^{38}\cdot 3^{10}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.97$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{45} a^{10} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{2}{15} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{2}{9}$, $\frac{1}{45} a^{11} - \frac{1}{45} a^{6} - \frac{1}{9} a^{5} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{135} a^{12} - \frac{1}{27} a^{9} - \frac{2}{45} a^{7} - \frac{2}{27} a^{6} + \frac{1}{9} a^{4} - \frac{2}{27} a^{3} - \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{10}{27}$, $\frac{1}{135} a^{13} + \frac{1}{135} a^{10} - \frac{2}{45} a^{8} + \frac{4}{27} a^{7} + \frac{1}{9} a^{6} - \frac{7}{45} a^{5} + \frac{4}{27} a^{4} + \frac{2}{9} a^{3} + \frac{4}{9} a^{2} - \frac{10}{27} a + \frac{2}{9}$, $\frac{1}{135} a^{14} + \frac{1}{135} a^{11} - \frac{2}{45} a^{9} + \frac{1}{27} a^{8} - \frac{1}{9} a^{7} - \frac{2}{45} a^{6} - \frac{2}{27} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{7}{27} a^{2} + \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{135} a^{15} - \frac{1}{27} a^{9} - \frac{1}{9} a^{6} + \frac{1}{15} a^{5} + \frac{13}{27} a^{3} + \frac{1}{3} a + \frac{7}{27}$, $\frac{1}{2025} a^{16} + \frac{7}{2025} a^{15} - \frac{7}{2025} a^{14} - \frac{2}{675} a^{13} + \frac{1}{405} a^{12} - \frac{16}{2025} a^{11} - \frac{2}{2025} a^{10} - \frac{16}{675} a^{9} - \frac{44}{2025} a^{8} + \frac{2}{15} a^{7} + \frac{13}{81} a^{6} + \frac{23}{405} a^{5} - \frac{38}{405} a^{4} + \frac{7}{45} a^{3} + \frac{35}{81} a^{2} + \frac{8}{81} a + \frac{37}{81}$, $\frac{1}{2025} a^{17} + \frac{4}{2025} a^{15} - \frac{2}{2025} a^{14} + \frac{2}{2025} a^{13} - \frac{2}{675} a^{12} + \frac{4}{405} a^{11} + \frac{11}{2025} a^{10} + \frac{37}{2025} a^{9} - \frac{52}{2025} a^{8} - \frac{52}{405} a^{7} - \frac{2}{15} a^{6} - \frac{2}{81} a^{5} - \frac{31}{405} a^{4} + \frac{19}{405} a^{3} + \frac{2}{27} a^{2} - \frac{10}{81} a + \frac{5}{81}$, $\frac{1}{25677298093079475} a^{18} - \frac{1}{2853033121453275} a^{17} - \frac{2760741266878}{25677298093079475} a^{16} + \frac{22085930135228}{25677298093079475} a^{15} + \frac{73213801409584}{25677298093079475} a^{14} + \frac{5778961472237}{2853033121453275} a^{13} - \frac{63858408377516}{25677298093079475} a^{12} - \frac{48887053218647}{25677298093079475} a^{11} + \frac{256165819324412}{25677298093079475} a^{10} - \frac{639998813304214}{25677298093079475} a^{9} - \frac{1375475340534124}{25677298093079475} a^{8} - \frac{211644303652676}{1711819872871965} a^{7} + \frac{802577516171179}{5135459618615895} a^{6} + \frac{784909480873408}{5135459618615895} a^{5} - \frac{775007929960501}{5135459618615895} a^{4} - \frac{560268056162764}{1711819872871965} a^{3} + \frac{216591646063315}{1027091923723179} a^{2} + \frac{77079239273659}{1027091923723179} a - \frac{212880474491000}{1027091923723179}$, $\frac{1}{12858908434735177205775} a^{19} + \frac{50077}{2571781686947035441155} a^{18} + \frac{2942070354393509674}{12858908434735177205775} a^{17} - \frac{11057758667099479}{476255867953154711325} a^{16} - \frac{4662220988255658679}{2571781686947035441155} a^{15} + \frac{33110400802277854861}{12858908434735177205775} a^{14} - \frac{10715158330762339834}{12858908434735177205775} a^{13} + \frac{1382248138372592497}{4286302811578392401925} a^{12} + \frac{36501603102492965363}{12858908434735177205775} a^{11} + \frac{16567696347881607713}{2571781686947035441155} a^{10} + \frac{449792790177043107653}{12858908434735177205775} a^{9} - \frac{9749656991543271154}{676784654459746168725} a^{8} + \frac{6942402174374811769}{514356337389407088231} a^{7} + \frac{21643604128568437622}{285753520771892826795} a^{6} - \frac{84267722161021264714}{2571781686947035441155} a^{5} + \frac{166517212665534963293}{2571781686947035441155} a^{4} - \frac{493033789861229760773}{2571781686947035441155} a^{3} - \frac{20124908868344652982}{171452112463135696077} a^{2} + \frac{16733884768042972816}{514356337389407088231} a - \frac{23983865177859649942}{171452112463135696077}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 14476205.398747573 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{5}, \sqrt{-6})\), 5.1.50000.1, 10.2.12500000000.1, 10.0.1244160000000000.16, 10.0.6220800000000000.8

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.4.2.1$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}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$