↑

Properties

Label 20.20.7668986217...4432.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 53^{15}$
Root discriminant $39.29$
Ramified primes $2, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_5$ (as 20T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -128, -356, 3704, 4637, -21178, -25550, 40496, 53513, -31240, -50258, 9162, 23428, -38, -5626, -416, 689, 56, -42, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 42*x^18 + 56*x^17 + 689*x^16 - 416*x^15 - 5626*x^14 - 38*x^13 + 23428*x^12 + 9162*x^11 - 50258*x^10 - 31240*x^9 + 53513*x^8 + 40496*x^7 - 25550*x^6 - 21178*x^5 + 4637*x^4 + 3704*x^3 - 356*x^2 - 128*x - 4)
 
gp: K = bnfinit(x^20 - 2*x^19 - 42*x^18 + 56*x^17 + 689*x^16 - 416*x^15 - 5626*x^14 - 38*x^13 + 23428*x^12 + 9162*x^11 - 50258*x^10 - 31240*x^9 + 53513*x^8 + 40496*x^7 - 25550*x^6 - 21178*x^5 + 4637*x^4 + 3704*x^3 - 356*x^2 - 128*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 42 x^{18} + 56 x^{17} + 689 x^{16} - 416 x^{15} - 5626 x^{14} - 38 x^{13} + 23428 x^{12} + 9162 x^{11} - 50258 x^{10} - 31240 x^{9} + 53513 x^{8} + 40496 x^{7} - 25550 x^{6} - 21178 x^{5} + 4637 x^{4} + 3704 x^{3} - 356 x^{2} - 128 x - 4 \)

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:  \(76689862179190073956224893714432=2^{20}\cdot 53^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.29$
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:  $2, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{7}{16} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{160} a^{17} - \frac{3}{160} a^{16} - \frac{3}{80} a^{15} - \frac{1}{8} a^{14} - \frac{9}{80} a^{12} - \frac{7}{40} a^{11} + \frac{9}{40} a^{10} + \frac{1}{80} a^{9} - \frac{1}{20} a^{8} + \frac{17}{40} a^{7} - \frac{29}{80} a^{6} - \frac{1}{160} a^{5} + \frac{59}{160} a^{4} - \frac{17}{40} a^{3} - \frac{1}{8} a^{2} + \frac{19}{40} a + \frac{13}{40}$, $\frac{1}{1600} a^{18} + \frac{1}{400} a^{17} + \frac{23}{1600} a^{16} + \frac{79}{800} a^{15} + \frac{9}{80} a^{14} - \frac{29}{800} a^{13} - \frac{97}{800} a^{12} - \frac{19}{80} a^{11} - \frac{3}{800} a^{10} + \frac{173}{800} a^{9} - \frac{7}{400} a^{8} - \frac{211}{800} a^{7} + \frac{233}{1600} a^{6} - \frac{13}{100} a^{5} + \frac{151}{320} a^{4} - \frac{9}{25} a^{3} + \frac{67}{200} a^{2} - \frac{87}{200} a - \frac{39}{400}$, $\frac{1}{415299758805872000} a^{19} + \frac{84913409488131}{415299758805872000} a^{18} + \frac{780252138062531}{415299758805872000} a^{17} + \frac{3946326466566779}{415299758805872000} a^{16} + \frac{23566209617446623}{207649879402936000} a^{15} - \frac{24015373384163999}{207649879402936000} a^{14} + \frac{74887188001691}{10382493970146800} a^{13} - \frac{10622414533905309}{207649879402936000} a^{12} + \frac{31231190957116567}{207649879402936000} a^{11} - \frac{11718393769377627}{51912469850734000} a^{10} + \frac{6090846383193857}{207649879402936000} a^{9} + \frac{20141771033752811}{207649879402936000} a^{8} + \frac{2137677736323937}{8836165080976000} a^{7} + \frac{188193706037095383}{415299758805872000} a^{6} + \frac{142958274561737339}{415299758805872000} a^{5} - \frac{64800521640265791}{415299758805872000} a^{4} + \frac{9221835250629523}{51912469850734000} a^{3} - \frac{3828728900928007}{12978117462683500} a^{2} + \frac{21545862857737063}{103824939701468000} a - \frac{43750782539926253}{103824939701468000}$

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:  \( 8458588550.38 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.2382032.1, 5.5.2382032.1 x5, 10.10.300726051798272.1 x5

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

Sibling fields

Degree 5 sibling: 5.5.2382032.1
Degree 10 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$