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Properties

Label 20.4.15418170861...0000.9
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{16}$
Root discriminant $28.80$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![199, 302, -990, -1134, 2174, 1968, 542, -8836, 7589, 2298, -7998, 5804, -1026, -1394, 1328, -618, 180, -46, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 14*x^18 - 46*x^17 + 180*x^16 - 618*x^15 + 1328*x^14 - 1394*x^13 - 1026*x^12 + 5804*x^11 - 7998*x^10 + 2298*x^9 + 7589*x^8 - 8836*x^7 + 542*x^6 + 1968*x^5 + 2174*x^4 - 1134*x^3 - 990*x^2 + 302*x + 199)
 
gp: K = bnfinit(x^20 - 4*x^19 + 14*x^18 - 46*x^17 + 180*x^16 - 618*x^15 + 1328*x^14 - 1394*x^13 - 1026*x^12 + 5804*x^11 - 7998*x^10 + 2298*x^9 + 7589*x^8 - 8836*x^7 + 542*x^6 + 1968*x^5 + 2174*x^4 - 1134*x^3 - 990*x^2 + 302*x + 199, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 14 x^{18} - 46 x^{17} + 180 x^{16} - 618 x^{15} + 1328 x^{14} - 1394 x^{13} - 1026 x^{12} + 5804 x^{11} - 7998 x^{10} + 2298 x^{9} + 7589 x^{8} - 8836 x^{7} + 542 x^{6} + 1968 x^{5} + 2174 x^{4} - 1134 x^{3} - 990 x^{2} + 302 x + 199 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.80$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{4}{11} a^{14} + \frac{5}{11} a^{13} - \frac{1}{11} a^{12} + \frac{4}{11} a^{11} - \frac{2}{11} a^{10} + \frac{2}{11} a^{9} + \frac{3}{11} a^{8} + \frac{5}{11} a^{7} - \frac{4}{11} a^{6} + \frac{5}{11} a^{5} - \frac{2}{11} a^{4} - \frac{4}{11} a^{3} + \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{1}{11} a^{13} - \frac{3}{11} a^{12} + \frac{4}{11} a^{11} - \frac{1}{11} a^{10} - \frac{5}{11} a^{9} + \frac{4}{11} a^{8} - \frac{2}{11} a^{7} - \frac{1}{11} a^{6} + \frac{4}{11} a^{4} + \frac{5}{11} a^{3} + \frac{4}{11} a^{2} - \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{17} + \frac{1}{11} a^{14} - \frac{3}{11} a^{13} + \frac{4}{11} a^{12} - \frac{1}{11} a^{11} - \frac{5}{11} a^{10} + \frac{4}{11} a^{9} - \frac{2}{11} a^{8} - \frac{1}{11} a^{7} + \frac{4}{11} a^{5} + \frac{5}{11} a^{4} + \frac{4}{11} a^{3} - \frac{4}{11} a^{2} - \frac{4}{11} a$, $\frac{1}{11} a^{18} + \frac{4}{11} a^{14} - \frac{1}{11} a^{13} + \frac{2}{11} a^{11} - \frac{5}{11} a^{10} - \frac{4}{11} a^{9} - \frac{4}{11} a^{8} - \frac{5}{11} a^{7} - \frac{3}{11} a^{6} - \frac{5}{11} a^{4} - \frac{4}{11} a^{2} - \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{358821159310189124093537926018422271} a^{19} + \frac{829490214760693647463059410680009}{32620105391835374917594356910765661} a^{18} - \frac{11941528601627134922264709609174913}{358821159310189124093537926018422271} a^{17} - \frac{12216191258734305181575636159618994}{358821159310189124093537926018422271} a^{16} + \frac{7022009429829912846905826295530912}{358821159310189124093537926018422271} a^{15} + \frac{172870319041154378183460720841882241}{358821159310189124093537926018422271} a^{14} - \frac{119340217729533413365290598151851338}{358821159310189124093537926018422271} a^{13} - \frac{91065638214427383797877531115613696}{358821159310189124093537926018422271} a^{12} - \frac{135067941145029449580045189127802993}{358821159310189124093537926018422271} a^{11} + \frac{14263645569409611949349617430072957}{32620105391835374917594356910765661} a^{10} - \frac{128910790996000904794053734316877633}{358821159310189124093537926018422271} a^{9} + \frac{38495847657473143026428659083390840}{358821159310189124093537926018422271} a^{8} - \frac{177032008167521290335650810826221177}{358821159310189124093537926018422271} a^{7} - \frac{99932244681105239768898867477563144}{358821159310189124093537926018422271} a^{6} - \frac{11932419615057885512401195829364198}{358821159310189124093537926018422271} a^{5} + \frac{49388946450117730940633961278063241}{358821159310189124093537926018422271} a^{4} - \frac{524704675407201424902882004878818}{358821159310189124093537926018422271} a^{3} - \frac{115559565261998589233478928368850751}{358821159310189124093537926018422271} a^{2} - \frac{160311307827062374322447286613489630}{358821159310189124093537926018422271} a + \frac{55829855828175063013198175348285894}{358821159310189124093537926018422271}$

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:  $11$
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:  \( 4314990.02416 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.219503494144.2

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 R $20$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$