↑

Properties

Label 20.20.9534886571...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{18}\cdot 5^{15}\cdot 73^{8}$
Root discriminant $50.00$
Ramified primes $3, 5, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times A_5$ (as 20T63)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 153, -936, 2106, 1314, -12177, 9267, 21255, -28776, -13239, 32154, -780, -16775, 4129, 4054, -1578, -368, 218, -3, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 - 3*x^18 + 218*x^17 - 368*x^16 - 1578*x^15 + 4054*x^14 + 4129*x^13 - 16775*x^12 - 780*x^11 + 32154*x^10 - 13239*x^9 - 28776*x^8 + 21255*x^7 + 9267*x^6 - 12177*x^5 + 1314*x^4 + 2106*x^3 - 936*x^2 + 153*x - 9)
 
gp: K = bnfinit(x^20 - 9*x^19 - 3*x^18 + 218*x^17 - 368*x^16 - 1578*x^15 + 4054*x^14 + 4129*x^13 - 16775*x^12 - 780*x^11 + 32154*x^10 - 13239*x^9 - 28776*x^8 + 21255*x^7 + 9267*x^6 - 12177*x^5 + 1314*x^4 + 2106*x^3 - 936*x^2 + 153*x - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} - 3 x^{18} + 218 x^{17} - 368 x^{16} - 1578 x^{15} + 4054 x^{14} + 4129 x^{13} - 16775 x^{12} - 780 x^{11} + 32154 x^{10} - 13239 x^{9} - 28776 x^{8} + 21255 x^{7} + 9267 x^{6} - 12177 x^{5} + 1314 x^{4} + 2106 x^{3} - 936 x^{2} + 153 x - 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9534886571062921057910430908203125=3^{18}\cdot 5^{15}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.00$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6}$, $\frac{1}{62828431308990507} a^{19} - \frac{2208816955670400}{20942810436330169} a^{18} + \frac{1933388233280199}{20942810436330169} a^{17} + \frac{2072954229812579}{20942810436330169} a^{16} - \frac{77610712755359}{2026723590612597} a^{15} - \frac{459854286355400}{20942810436330169} a^{14} - \frac{403435401780410}{20942810436330169} a^{13} - \frac{4007583013452274}{62828431308990507} a^{12} - \frac{30532760819257676}{62828431308990507} a^{11} + \frac{9309999756449794}{62828431308990507} a^{10} + \frac{9754103551320409}{62828431308990507} a^{9} + \frac{5201346854580997}{62828431308990507} a^{8} + \frac{7669593330449076}{20942810436330169} a^{7} + \frac{5206859105470628}{20942810436330169} a^{6} - \frac{4824700847446216}{20942810436330169} a^{5} - \frac{5121156083097155}{20942810436330169} a^{4} + \frac{9739123734153017}{20942810436330169} a^{3} - \frac{8554694902815759}{20942810436330169} a^{2} + \frac{9159439315052821}{20942810436330169} a + \frac{9497010408272945}{20942810436330169}$

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:  $19$
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:  \( 49407493166.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times A_5$ (as 20T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 20 conjugacy class representatives for $C_4\times A_5$
Character table for $C_4\times A_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.10791225.1, 10.10.582252685003125.1

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

Sibling fields

Degree 24 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 $20$ R R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.14.3$x^{12} - 12 x^{11} + 3 x^{10} - 9 x^{7} - 6 x^{6} - 9 x^{3} + 9 x^{2} - 9$$6$$2$$14$$S_3 \times C_4$$[3/2]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$73$73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.12.8.2$x^{12} - 389017 x^{3} + 369177133$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$