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Properties

Label 20.0.10973285720...9849.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 65657^{4}$
Root discriminant $15.92$
Ramified primes $3, 65657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T279

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 5, -8, 20, -29, 39, -56, 58, -60, 71, -60, 58, -56, 39, -29, 20, -8, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 8*x^17 + 20*x^16 - 29*x^15 + 39*x^14 - 56*x^13 + 58*x^12 - 60*x^11 + 71*x^10 - 60*x^9 + 58*x^8 - 56*x^7 + 39*x^6 - 29*x^5 + 20*x^4 - 8*x^3 + 5*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 5*x^18 - 8*x^17 + 20*x^16 - 29*x^15 + 39*x^14 - 56*x^13 + 58*x^12 - 60*x^11 + 71*x^10 - 60*x^9 + 58*x^8 - 56*x^7 + 39*x^6 - 29*x^5 + 20*x^4 - 8*x^3 + 5*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 5 x^{18} - 8 x^{17} + 20 x^{16} - 29 x^{15} + 39 x^{14} - 56 x^{13} + 58 x^{12} - 60 x^{11} + 71 x^{10} - 60 x^{9} + 58 x^{8} - 56 x^{7} + 39 x^{6} - 29 x^{5} + 20 x^{4} - 8 x^{3} + 5 x^{2} - x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1097328572072844211779849=3^{10}\cdot 65657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.92$
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:  $3, 65657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1047} a^{18} + \frac{34}{1047} a^{17} + \frac{49}{349} a^{16} - \frac{44}{349} a^{15} + \frac{488}{1047} a^{14} + \frac{431}{1047} a^{13} - \frac{22}{1047} a^{12} - \frac{70}{349} a^{11} + \frac{59}{1047} a^{10} + \frac{121}{1047} a^{9} + \frac{59}{1047} a^{8} - \frac{70}{349} a^{7} - \frac{22}{1047} a^{6} + \frac{431}{1047} a^{5} + \frac{488}{1047} a^{4} - \frac{44}{349} a^{3} + \frac{49}{349} a^{2} + \frac{34}{1047} a + \frac{1}{1047}$, $\frac{1}{26175} a^{19} - \frac{2}{26175} a^{18} - \frac{1406}{8725} a^{17} - \frac{571}{1745} a^{16} - \frac{2093}{5235} a^{15} - \frac{7714}{26175} a^{14} - \frac{12397}{26175} a^{13} - \frac{853}{8725} a^{12} - \frac{9133}{26175} a^{11} + \frac{6373}{26175} a^{10} + \frac{6173}{26175} a^{9} + \frac{2014}{8725} a^{8} + \frac{6491}{26175} a^{7} - \frac{9247}{26175} a^{6} - \frac{2464}{26175} a^{5} - \frac{831}{1745} a^{4} - \frac{511}{1745} a^{3} - \frac{10493}{26175} a^{2} - \frac{8552}{26175} a - \frac{1408}{8725}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{25894}{26175} a^{19} - \frac{15896}{8725} a^{18} + \frac{120608}{26175} a^{17} - \frac{17494}{1745} a^{16} + \frac{108208}{5235} a^{15} - \frac{294447}{8725} a^{14} + \frac{328044}{8725} a^{13} - \frac{1255721}{26175} a^{12} + \frac{1495973}{26175} a^{11} - \frac{1287163}{26175} a^{10} + \frac{1405162}{26175} a^{9} - \frac{1534427}{26175} a^{8} + \frac{1136579}{26175} a^{7} - \frac{333081}{8725} a^{6} + \frac{278878}{8725} a^{5} - \frac{87652}{5235} a^{4} + \frac{16771}{1745} a^{3} - \frac{165617}{26175} a^{2} + \frac{18629}{8725} a - \frac{13006}{26175} \) (order $6$)
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:  \( 10958.1798553 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T279:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.65657.1, 10.0.1047534520707.1, 10.6.349178173569.1, 10.4.12932524947.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
65657Data not computed