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Properties

Label 20.2.18079489015...9712.1
Degree $20$
Signature $[2, 9]$
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![463, -641, 519, -777, 984, -548, -199, 1108, -1759, 1833, -1527, 1368, -1033, 528, -153, 55, -26, -4, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 4*x^17 - 26*x^16 + 55*x^15 - 153*x^14 + 528*x^13 - 1033*x^12 + 1368*x^11 - 1527*x^10 + 1833*x^9 - 1759*x^8 + 1108*x^7 - 199*x^6 - 548*x^5 + 984*x^4 - 777*x^3 + 519*x^2 - 641*x + 463)
 
gp: K = bnfinit(x^20 - 6*x^19 + 13*x^18 - 4*x^17 - 26*x^16 + 55*x^15 - 153*x^14 + 528*x^13 - 1033*x^12 + 1368*x^11 - 1527*x^10 + 1833*x^9 - 1759*x^8 + 1108*x^7 - 199*x^6 - 548*x^5 + 984*x^4 - 777*x^3 + 519*x^2 - 641*x + 463, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 13 x^{18} - 4 x^{17} - 26 x^{16} + 55 x^{15} - 153 x^{14} + 528 x^{13} - 1033 x^{12} + 1368 x^{11} - 1527 x^{10} + 1833 x^{9} - 1759 x^{8} + 1108 x^{7} - 199 x^{6} - 548 x^{5} + 984 x^{4} - 777 x^{3} + 519 x^{2} - 641 x + 463 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{178} a^{18} - \frac{7}{89} a^{17} + \frac{10}{89} a^{16} - \frac{29}{178} a^{15} - \frac{25}{178} a^{14} + \frac{7}{178} a^{13} + \frac{13}{178} a^{12} + \frac{45}{178} a^{11} + \frac{1}{178} a^{10} - \frac{36}{89} a^{9} - \frac{77}{178} a^{8} - \frac{24}{89} a^{7} - \frac{27}{89} a^{6} - \frac{3}{89} a^{5} - \frac{44}{89} a^{4} - \frac{15}{178} a^{3} + \frac{10}{89} a^{2} - \frac{37}{89} a - \frac{5}{89}$, $\frac{1}{2269020134736469129122687020978} a^{19} + \frac{2188572704554593787240285233}{2269020134736469129122687020978} a^{18} - \frac{472947206629988339566666160813}{2269020134736469129122687020978} a^{17} - \frac{41057139208588190857646667426}{1134510067368234564561343510489} a^{16} + \frac{392155960698430089340265679111}{2269020134736469129122687020978} a^{15} + \frac{307432118173325729651768828539}{2269020134736469129122687020978} a^{14} + \frac{418501913790946208802295408544}{1134510067368234564561343510489} a^{13} - \frac{408651901612127361971679065873}{2269020134736469129122687020978} a^{12} + \frac{789870089019978241960070612475}{2269020134736469129122687020978} a^{11} - \frac{486805838252551142895263437353}{1134510067368234564561343510489} a^{10} - \frac{382526668148109040345293920092}{1134510067368234564561343510489} a^{9} + \frac{2124284495724413400564614215}{25494608255465945271041427202} a^{8} + \frac{19802730322967935856855980333}{1134510067368234564561343510489} a^{7} - \frac{220769605355408416519108954401}{2269020134736469129122687020978} a^{6} + \frac{176951368160941294786627757389}{2269020134736469129122687020978} a^{5} + \frac{361454307665747137658720921937}{2269020134736469129122687020978} a^{4} + \frac{605640406724460669714499985945}{2269020134736469129122687020978} a^{3} + \frac{675715480558331416684967027753}{2269020134736469129122687020978} a^{2} - \frac{1065129173339229264753689862279}{2269020134736469129122687020978} a - \frac{428120641733997835728710570937}{2269020134736469129122687020978}$

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:  $10$
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:  \( 839071.478682 \) (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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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