↑

Properties

Label 20.0.27134199920...2841.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{14}\cdot 97^{2}\cdot 1567^{4}$
Root discriminant $14.85$
Ramified primes $3, 97, 1567$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 7, 8, 21, 22, 33, 29, 30, 13, 15, -13, 30, -29, 33, -22, 21, -8, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 7*x^18 - 8*x^17 + 21*x^16 - 22*x^15 + 33*x^14 - 29*x^13 + 30*x^12 - 13*x^11 + 15*x^10 + 13*x^9 + 30*x^8 + 29*x^7 + 33*x^6 + 22*x^5 + 21*x^4 + 8*x^3 + 7*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 7*x^18 - 8*x^17 + 21*x^16 - 22*x^15 + 33*x^14 - 29*x^13 + 30*x^12 - 13*x^11 + 15*x^10 + 13*x^9 + 30*x^8 + 29*x^7 + 33*x^6 + 22*x^5 + 21*x^4 + 8*x^3 + 7*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 7 x^{18} - 8 x^{17} + 21 x^{16} - 22 x^{15} + 33 x^{14} - 29 x^{13} + 30 x^{12} - 13 x^{11} + 15 x^{10} + 13 x^{9} + 30 x^{8} + 29 x^{7} + 33 x^{6} + 22 x^{5} + 21 x^{4} + 8 x^{3} + 7 x^{2} + 2 x + 1 \)

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:  \(271341999200397872102841=3^{14}\cdot 97^{2}\cdot 1567^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.85$
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:  $3, 97, 1567$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{95517} a^{18} - \frac{6008}{95517} a^{17} - \frac{16}{10613} a^{16} - \frac{3806}{31839} a^{15} - \frac{1607}{31839} a^{14} + \frac{1835}{95517} a^{13} - \frac{1056}{10613} a^{12} + \frac{37955}{95517} a^{11} - \frac{10601}{31839} a^{10} + \frac{2597}{10613} a^{9} + \frac{10601}{31839} a^{8} - \frac{15110}{95517} a^{7} + \frac{1056}{10613} a^{6} + \frac{44287}{95517} a^{5} + \frac{12220}{31839} a^{4} - \frac{14419}{31839} a^{3} - \frac{10565}{31839} a^{2} - \frac{16621}{95517} a + \frac{10612}{95517}$, $\frac{1}{95517} a^{19} + \frac{9218}{95517} a^{17} + \frac{1658}{10613} a^{16} + \frac{2960}{31839} a^{15} + \frac{10757}{95517} a^{14} - \frac{1118}{95517} a^{13} - \frac{6589}{95517} a^{12} - \frac{29081}{95517} a^{11} - \frac{5210}{10613} a^{10} + \frac{15599}{31839} a^{9} - \frac{40364}{95517} a^{8} - \frac{30226}{95517} a^{7} - \frac{38525}{95517} a^{6} - \frac{29245}{95517} a^{5} - \frac{2260}{10613} a^{4} + \frac{5076}{10613} a^{3} + \frac{20717}{95517} a^{2} - \frac{1252}{95517} a + \frac{47057}{95517}$

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:  \( \frac{6356}{10613} a^{19} - \frac{34996}{31839} a^{18} + \frac{41833}{10613} a^{17} - \frac{43647}{10613} a^{16} + \frac{369758}{31839} a^{15} - \frac{362962}{31839} a^{14} + \frac{546359}{31839} a^{13} - \frac{478766}{31839} a^{12} + \frac{487162}{31839} a^{11} - \frac{194278}{31839} a^{10} + \frac{91445}{10613} a^{9} + \frac{235016}{31839} a^{8} + \frac{612599}{31839} a^{7} + \frac{577246}{31839} a^{6} + \frac{622969}{31839} a^{5} + \frac{347726}{31839} a^{4} + \frac{300127}{31839} a^{3} + \frac{20033}{10613} a^{2} + \frac{27410}{10613} a + \frac{20561}{31839} \) (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:  \( 6529.50946942 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.3.14103.1, 10.6.173634993657.1, 10.0.520904980971.1, 10.0.596683827.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
1567Data not computed