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Properties

Label 20.0.66719722723...8481.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 13^{2}\cdot 401^{8}$
Root discriminant $24.62$
Ramified primes $3, 13, 401$
Class number $4$
Class group $[2, 2]$
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -3, 70, -79, 519, -385, 522, -69, 280, 27, 72, 141, 42, -29, 105, -48, 42, -16, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 11*x^18 - 16*x^17 + 42*x^16 - 48*x^15 + 105*x^14 - 29*x^13 + 42*x^12 + 141*x^11 + 72*x^10 + 27*x^9 + 280*x^8 - 69*x^7 + 522*x^6 - 385*x^5 + 519*x^4 - 79*x^3 + 70*x^2 - 3*x + 9)
 
gp: K = bnfinit(x^20 - 3*x^19 + 11*x^18 - 16*x^17 + 42*x^16 - 48*x^15 + 105*x^14 - 29*x^13 + 42*x^12 + 141*x^11 + 72*x^10 + 27*x^9 + 280*x^8 - 69*x^7 + 522*x^6 - 385*x^5 + 519*x^4 - 79*x^3 + 70*x^2 - 3*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 11 x^{18} - 16 x^{17} + 42 x^{16} - 48 x^{15} + 105 x^{14} - 29 x^{13} + 42 x^{12} + 141 x^{11} + 72 x^{10} + 27 x^{9} + 280 x^{8} - 69 x^{7} + 522 x^{6} - 385 x^{5} + 519 x^{4} - 79 x^{3} + 70 x^{2} - 3 x + 9 \)

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:  \(6671972272300106674226558481=3^{10}\cdot 13^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.62$
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:  $3, 13, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{83} a^{17} + \frac{2}{83} a^{16} - \frac{33}{83} a^{15} - \frac{14}{83} a^{14} - \frac{20}{83} a^{13} - \frac{1}{83} a^{12} - \frac{14}{83} a^{11} + \frac{16}{83} a^{10} + \frac{22}{83} a^{9} + \frac{19}{83} a^{8} - \frac{24}{83} a^{7} + \frac{34}{83} a^{6} - \frac{1}{83} a^{5} + \frac{39}{83} a^{4} - \frac{5}{83} a^{3} + \frac{31}{83} a^{2} - \frac{34}{83} a + \frac{22}{83}$, $\frac{1}{83} a^{18} - \frac{37}{83} a^{16} - \frac{31}{83} a^{15} + \frac{8}{83} a^{14} + \frac{39}{83} a^{13} - \frac{12}{83} a^{12} - \frac{39}{83} a^{11} - \frac{10}{83} a^{10} - \frac{25}{83} a^{9} + \frac{21}{83} a^{8} - \frac{1}{83} a^{7} + \frac{14}{83} a^{6} + \frac{41}{83} a^{5} + \frac{41}{83} a^{3} - \frac{13}{83} a^{2} + \frac{7}{83} a + \frac{39}{83}$, $\frac{1}{1010219511430180599} a^{19} + \frac{335611869792415}{336739837143393533} a^{18} - \frac{845732865089077}{1010219511430180599} a^{17} + \frac{111293397816760910}{1010219511430180599} a^{16} + \frac{45675112059638270}{336739837143393533} a^{15} - \frac{165936187273187234}{336739837143393533} a^{14} + \frac{10768866625245915}{336739837143393533} a^{13} - \frac{297698822181952763}{1010219511430180599} a^{12} + \frac{80953167209270779}{336739837143393533} a^{11} - \frac{2256197804233658}{336739837143393533} a^{10} - \frac{1819196180242781}{336739837143393533} a^{9} + \frac{99597604785818712}{336739837143393533} a^{8} + \frac{207303094103687359}{1010219511430180599} a^{7} + \frac{80773525841433780}{336739837143393533} a^{6} - \frac{58694924974909483}{336739837143393533} a^{5} + \frac{319048679314378088}{1010219511430180599} a^{4} + \frac{102560379743154781}{336739837143393533} a^{3} - \frac{324427952273436202}{1010219511430180599} a^{2} + \frac{2433997495911670}{1010219511430180599} a - \frac{141638846595072323}{336739837143393533}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{2512377809661056}{1010219511430180599} a^{19} + \frac{2038987998243536}{336739837143393533} a^{18} - \frac{13351220705202509}{1010219511430180599} a^{17} + \frac{113055451409372800}{1010219511430180599} a^{16} - \frac{38365841749934421}{336739837143393533} a^{15} + \frac{156860439698150215}{336739837143393533} a^{14} - \frac{126832496346244627}{336739837143393533} a^{13} + \frac{1431665706398973335}{1010219511430180599} a^{12} - \frac{76317893110289970}{336739837143393533} a^{11} + \frac{364373769729570627}{336739837143393533} a^{10} + \frac{825856762010777361}{336739837143393533} a^{9} + \frac{449718343255021219}{336739837143393533} a^{8} + \frac{1550518355873677352}{1010219511430180599} a^{7} + \frac{1488968438310308590}{336739837143393533} a^{6} + \frac{395685455516436049}{336739837143393533} a^{5} + \frac{6564907402591659715}{1010219511430180599} a^{4} - \frac{1152774864943253612}{336739837143393533} a^{3} + \frac{7883838087950646562}{1010219511430180599} a^{2} - \frac{410061779212973272}{1010219511430180599} a + \frac{367932817956517545}{336739837143393533} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 553903.652517 \)
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, 10.6.336140500813.1, 10.4.81682141697559.1, 10.0.6283241669043.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{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}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
401Data not computed