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Properties

Label 20.0.67955065915...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $438.14$
Ramified primes $2, 3, 5, 41$
Class number $25434258432$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 397410288]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1329697265625, 0, 9810210937500, 0, 9074116796875, 0, 3135590312500, 0, 491561171875, 0, 37375600000, 0, 1372885000, 0, 24743500, 0, 214225, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 214225*x^16 + 24743500*x^14 + 1372885000*x^12 + 37375600000*x^10 + 491561171875*x^8 + 3135590312500*x^6 + 9074116796875*x^4 + 9810210937500*x^2 + 1329697265625)
 
gp: K = bnfinit(x^20 + 820*x^18 + 214225*x^16 + 24743500*x^14 + 1372885000*x^12 + 37375600000*x^10 + 491561171875*x^8 + 3135590312500*x^6 + 9074116796875*x^4 + 9810210937500*x^2 + 1329697265625, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 214225 x^{16} + 24743500 x^{14} + 1372885000 x^{12} + 37375600000 x^{10} + 491561171875 x^{8} + 3135590312500 x^{6} + 9074116796875 x^{4} + 9810210937500 x^{2} + 1329697265625 \)

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:  \(67955065915961530473358856164812748971048960000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $438.14$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(4099,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(269,·)$, $\chi_{4920}(1229,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(3911,·)$, $\chi_{4920}(3139,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(4459,·)$, $\chi_{4920}(4511,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(4699,·)$, $\chi_{4920}(1829,·)$, $\chi_{4920}(551,·)$, $\chi_{4920}(619,·)$, $\chi_{4920}(2669,·)$, $\chi_{4920}(1589,·)$, $\chi_{4920}(4151,·)$, $\chi_{4920}(3071,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{75} a^{5} - \frac{1}{15} a^{3} + \frac{1}{3} a$, $\frac{1}{375} a^{6} - \frac{1}{75} a^{4} + \frac{1}{15} a^{2}$, $\frac{1}{375} a^{7} + \frac{1}{3} a$, $\frac{1}{5625} a^{8} + \frac{1}{75} a^{4} + \frac{4}{45} a^{2}$, $\frac{1}{5625} a^{9} - \frac{2}{45} a^{3} - \frac{1}{3} a$, $\frac{1}{1153125} a^{10} - \frac{4}{225} a^{4} + \frac{1}{15} a^{2}$, $\frac{1}{3459375} a^{11} - \frac{1}{16875} a^{9} + \frac{2}{675} a^{5} + \frac{8}{135} a^{3} - \frac{1}{3} a$, $\frac{1}{17296875} a^{12} + \frac{1}{3459375} a^{10} + \frac{2}{3375} a^{6} + \frac{2}{675} a^{4} + \frac{1}{15} a^{2}$, $\frac{1}{17296875} a^{13} + \frac{1}{16875} a^{9} + \frac{2}{3375} a^{7} + \frac{1}{135} a^{3} + \frac{1}{3} a$, $\frac{1}{86484375} a^{14} - \frac{1}{3459375} a^{10} - \frac{1}{16875} a^{8} - \frac{2}{675} a^{4} + \frac{2}{45} a^{2}$, $\frac{1}{86484375} a^{15} + \frac{1}{16875} a^{9} + \frac{8}{135} a^{3} + \frac{1}{3} a$, $\frac{1}{25773374211328125} a^{16} - \frac{7241731}{1718224947421875} a^{14} + \frac{238576}{68728997896875} a^{12} + \frac{300364}{1649495949525} a^{10} - \frac{19156442}{335263404375} a^{8} - \frac{16663466}{67052680875} a^{6} - \frac{285516097}{40231608525} a^{4} - \frac{68686409}{894035745} a^{2} - \frac{8174539}{19867461}$, $\frac{1}{25773374211328125} a^{17} - \frac{7241731}{1718224947421875} a^{15} + \frac{238576}{68728997896875} a^{13} - \frac{22056883}{206186993690625} a^{11} + \frac{711019}{335263404375} a^{9} - \frac{16663466}{67052680875} a^{7} + \frac{131700584}{40231608525} a^{5} - \frac{820513}{298011915} a^{3} + \frac{1690145}{6622487} a$, $\frac{1}{1141631610690779296875} a^{18} - \frac{127}{45665264427631171875} a^{16} - \frac{1854565829}{608870192368415625} a^{14} - \frac{8269691689}{9133052885526234375} a^{12} + \frac{357231137294}{1826610577105246875} a^{10} - \frac{191697868333}{2970098499358125} a^{8} - \frac{1617346548778}{1782059099614875} a^{6} + \frac{1300967912674}{71282363984595} a^{4} + \frac{198621560791}{7920262664955} a^{2} + \frac{76225351823}{176005836999}$, $\frac{1}{1141631610690779296875} a^{19} - \frac{127}{45665264427631171875} a^{17} - \frac{1854565829}{608870192368415625} a^{15} - \frac{8269691689}{9133052885526234375} a^{13} - \frac{170786373703}{1826610577105246875} a^{11} - \frac{15692031334}{2970098499358125} a^{9} - \frac{1617346548778}{1782059099614875} a^{7} + \frac{696646942403}{356411819922975} a^{5} + \frac{257290173124}{7920262664955} a^{3} + \frac{76225351823}{176005836999} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{397410288}$, which has order $25434258432$ (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:  \( 1759608835.9021456 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1230}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{3}, \sqrt{-410})\), 5.5.2825761.1, 10.10.1986904914612636672.1, 10.0.8146310149911810355200000.1, 10.0.33523910081941606400000.1

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

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
5Data not computed
41Data not computed