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Properties

Label 20.6.18079489015...9712.3
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 727^{3}$
Root discriminant $25.87$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 128, -240, -388, -848, 111, 1240, 1550, 589, -466, -1089, -71, -210, 278, -119, 45, -23, 6, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 7*x^18 + 6*x^17 - 23*x^16 + 45*x^15 - 119*x^14 + 278*x^13 - 210*x^12 - 71*x^11 - 1089*x^10 - 466*x^9 + 589*x^8 + 1550*x^7 + 1240*x^6 + 111*x^5 - 848*x^4 - 388*x^3 - 240*x^2 + 128*x + 32)
 
gp: K = bnfinit(x^20 - 4*x^19 + 7*x^18 + 6*x^17 - 23*x^16 + 45*x^15 - 119*x^14 + 278*x^13 - 210*x^12 - 71*x^11 - 1089*x^10 - 466*x^9 + 589*x^8 + 1550*x^7 + 1240*x^6 + 111*x^5 - 848*x^4 - 388*x^3 - 240*x^2 + 128*x + 32, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 7 x^{18} + 6 x^{17} - 23 x^{16} + 45 x^{15} - 119 x^{14} + 278 x^{13} - 210 x^{12} - 71 x^{11} - 1089 x^{10} - 466 x^{9} + 589 x^{8} + 1550 x^{7} + 1240 x^{6} + 111 x^{5} - 848 x^{4} - 388 x^{3} - 240 x^{2} + 128 x + 32 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18079489015266793067734899712=-\,2^{10}\cdot 11^{16}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} + \frac{1}{8} a^{14} + \frac{1}{8} a^{13} + \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{9589022320831239930777221358352} a^{19} - \frac{50963583971752940085270819577}{4794511160415619965388610679176} a^{18} - \frac{668062252777532834744731683629}{9589022320831239930777221358352} a^{17} + \frac{538085967568987435775248881843}{2397255580207809982694305339588} a^{16} - \frac{1478422385261875025992037498431}{9589022320831239930777221358352} a^{15} + \frac{1647871833811690484792845727911}{9589022320831239930777221358352} a^{14} - \frac{1113096457632291661282475534993}{9589022320831239930777221358352} a^{13} + \frac{520760932528965636394334554775}{2397255580207809982694305339588} a^{12} + \frac{41196762234246935084434322767}{4794511160415619965388610679176} a^{11} - \frac{2872218689197868548923914448691}{9589022320831239930777221358352} a^{10} - \frac{1772667653739092835575405292719}{9589022320831239930777221358352} a^{9} - \frac{427221308937512788843085626535}{1198627790103904991347152669794} a^{8} + \frac{1366430864720654895137658273253}{9589022320831239930777221358352} a^{7} - \frac{531044006529776316462931971057}{1198627790103904991347152669794} a^{6} + \frac{405746567476131158217276698023}{1198627790103904991347152669794} a^{5} - \frac{2425118766917831386126776449689}{9589022320831239930777221358352} a^{4} + \frac{2069364123680063100288491204059}{4794511160415619965388610679176} a^{3} + \frac{189526450395305047981304231067}{599313895051952495673576334897} a^{2} + \frac{460795431908924703974935199181}{1198627790103904991347152669794} a + \frac{72974144374328962564463494528}{599313895051952495673576334897}$

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:  $12$
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:  \( 1869364.69729 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ $20$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
727Data not computed