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Properties

Label 20.4.61432542935...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 97^{2}\cdot 401^{8}$
Root discriminant $38.85$
Ramified primes $5, 97, 401$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -16, 43, 12, 55, 268, 246, -659, -1369, -1478, -222, 421, 622, 208, 47, -24, 40, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 4*x^18 + 40*x^17 - 24*x^16 + 47*x^15 + 208*x^14 + 622*x^13 + 421*x^12 - 222*x^11 - 1478*x^10 - 1369*x^9 - 659*x^8 + 246*x^7 + 268*x^6 + 55*x^5 + 12*x^4 + 43*x^3 - 16*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 4*x^18 + 40*x^17 - 24*x^16 + 47*x^15 + 208*x^14 + 622*x^13 + 421*x^12 - 222*x^11 - 1478*x^10 - 1369*x^9 - 659*x^8 + 246*x^7 + 268*x^6 + 55*x^5 + 12*x^4 + 43*x^3 - 16*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 4 x^{18} + 40 x^{17} - 24 x^{16} + 47 x^{15} + 208 x^{14} + 622 x^{13} + 421 x^{12} - 222 x^{11} - 1478 x^{10} - 1369 x^{9} - 659 x^{8} + 246 x^{7} + 268 x^{6} + 55 x^{5} + 12 x^{4} + 43 x^{3} - 16 x^{2} - x + 1 \)

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:  \(61432542935387226937873134765625=5^{10}\cdot 97^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.85$
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:  $5, 97, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{12} a^{15} + \frac{1}{12} a^{14} + \frac{1}{12} a^{13} + \frac{1}{6} a^{12} - \frac{5}{12} a^{11} + \frac{1}{3} a^{10} - \frac{1}{2} a^{9} + \frac{1}{12} a^{8} + \frac{1}{12} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{12} a - \frac{1}{6}$, $\frac{1}{24} a^{16} - \frac{1}{8} a^{14} + \frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{8} a^{11} - \frac{7}{24} a^{10} + \frac{1}{24} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{5}{24} a^{2} - \frac{7}{24} a - \frac{7}{24}$, $\frac{1}{24} a^{17} - \frac{1}{24} a^{15} + \frac{1}{6} a^{13} - \frac{5}{24} a^{12} + \frac{7}{24} a^{11} - \frac{3}{8} a^{10} + \frac{1}{4} a^{9} - \frac{5}{12} a^{8} - \frac{7}{24} a^{7} - \frac{1}{12} a^{6} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{24} a^{2} - \frac{3}{8} a - \frac{5}{12}$, $\frac{1}{144} a^{18} - \frac{1}{144} a^{17} - \frac{1}{144} a^{15} - \frac{13}{144} a^{14} + \frac{17}{144} a^{13} + \frac{1}{36} a^{12} + \frac{7}{16} a^{11} + \frac{1}{4} a^{10} - \frac{7}{16} a^{9} + \frac{55}{144} a^{8} + \frac{13}{48} a^{7} + \frac{13}{48} a^{6} + \frac{1}{4} a^{5} - \frac{17}{144} a^{4} - \frac{11}{72} a^{3} - \frac{47}{144} a^{2} + \frac{1}{8} a + \frac{55}{144}$, $\frac{1}{15061642694532473526864} a^{19} - \frac{1527297642755826181}{5020547564844157842288} a^{18} - \frac{5927361929315524183}{941352668408279595429} a^{17} - \frac{155647477479732961963}{15061642694532473526864} a^{16} + \frac{435503693962337452459}{15061642694532473526864} a^{15} - \frac{7832106608128914959}{1158587899579421040528} a^{14} + \frac{32876265313528274365}{193097983263236840088} a^{13} + \frac{1455213087637923378217}{15061642694532473526864} a^{12} - \frac{2550880001762628493}{2510273782422078921144} a^{11} - \frac{123722297768481469783}{1673515854948052614096} a^{10} + \frac{1265802122646706138171}{15061642694532473526864} a^{9} - \frac{2323319385285848051567}{15061642694532473526864} a^{8} - \frac{692945470965020701915}{5020547564844157842288} a^{7} - \frac{236760938762106405485}{627568445605519730286} a^{6} + \frac{1925393254202059664599}{15061642694532473526864} a^{5} + \frac{41547358042021752857}{104594740934253288381} a^{4} + \frac{2379797824925079606193}{5020547564844157842288} a^{3} + \frac{348899283503197247381}{1882705336816559190858} a^{2} - \frac{243464376819003383549}{15061642694532473526864} a + \frac{1157856045704103751693}{3765410673633118381716}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{9}$, which has order $9$ (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:  \( 12614808.3132 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.2.7837891485303125.1, 10.10.80803005003125.1, 10.2.2508125275297.1

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
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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed