↑

Properties

Label 20.0.31913046866...3841.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{16}\cdot 307^{2}\cdot 7369$
Root discriminant $18.85$
Ramified primes $11, 307, 7369$
Class number $1$
Class group Trivial
Galois group 20T846

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23, 20, 52, -15, 15, -42, 90, 42, 162, 80, 88, -29, 36, -12, 17, -6, 8, -3, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 6*x^15 + 17*x^14 - 12*x^13 + 36*x^12 - 29*x^11 + 88*x^10 + 80*x^9 + 162*x^8 + 42*x^7 + 90*x^6 - 42*x^5 + 15*x^4 - 15*x^3 + 52*x^2 + 20*x + 23)
 
gp: K = bnfinit(x^20 - x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 6*x^15 + 17*x^14 - 12*x^13 + 36*x^12 - 29*x^11 + 88*x^10 + 80*x^9 + 162*x^8 + 42*x^7 + 90*x^6 - 42*x^5 + 15*x^4 - 15*x^3 + 52*x^2 + 20*x + 23, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 3 x^{18} - 3 x^{17} + 8 x^{16} - 6 x^{15} + 17 x^{14} - 12 x^{13} + 36 x^{12} - 29 x^{11} + 88 x^{10} + 80 x^{9} + 162 x^{8} + 42 x^{7} + 90 x^{6} - 42 x^{5} + 15 x^{4} - 15 x^{3} + 52 x^{2} + 20 x + 23 \)

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:  \(31913046866560147064793841=11^{16}\cdot 307^{2}\cdot 7369\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.85$
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:  $11, 307, 7369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11} a^{8} + \frac{5}{11} a^{7} - \frac{1}{11} a^{6} + \frac{4}{11} a^{4} - \frac{3}{11} a^{3} + \frac{3}{11} a^{2} - \frac{3}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{9} - \frac{4}{11} a^{7} + \frac{5}{11} a^{6} + \frac{4}{11} a^{5} - \frac{1}{11} a^{4} - \frac{4}{11} a^{3} + \frac{4}{11} a^{2} + \frac{2}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{10} + \frac{3}{11} a^{7} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} + \frac{3}{11} a^{3} + \frac{3}{11} a^{2} - \frac{2}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{11} - \frac{4}{11} a^{7} + \frac{2}{11} a^{6} + \frac{1}{11} a^{5} + \frac{2}{11} a^{4} + \frac{1}{11} a^{3} + \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{12} - \frac{3}{11} a^{6} + \frac{2}{11} a^{5} - \frac{5}{11} a^{4} - \frac{1}{11} a^{3} + \frac{2}{11} a^{2} + \frac{5}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{7} + \frac{2}{11} a^{6} - \frac{5}{11} a^{5} - \frac{1}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2} + \frac{3}{11} a$, $\frac{1}{11} a^{14} - \frac{5}{11} a^{7} + \frac{3}{11} a^{6} - \frac{1}{11} a^{5} + \frac{3}{11} a^{4} - \frac{4}{11} a^{3} + \frac{1}{11} a^{2} + \frac{2}{11} a + \frac{5}{11}$, $\frac{1}{121} a^{15} - \frac{5}{121} a^{14} - \frac{1}{121} a^{13} + \frac{5}{121} a^{12} + \frac{1}{121} a^{10} - \frac{3}{121} a^{9} - \frac{2}{121} a^{8} + \frac{6}{121} a^{7} - \frac{29}{121} a^{6} + \frac{10}{121} a^{5} + \frac{17}{121} a^{4} + \frac{31}{121} a^{3} + \frac{24}{121} a^{2} - \frac{2}{11} a - \frac{43}{121}$, $\frac{1}{121} a^{16} - \frac{4}{121} a^{14} + \frac{3}{121} a^{12} + \frac{1}{121} a^{11} + \frac{2}{121} a^{10} + \frac{5}{121} a^{9} - \frac{4}{121} a^{8} + \frac{45}{121} a^{7} - \frac{14}{121} a^{6} - \frac{32}{121} a^{5} + \frac{28}{121} a^{4} + \frac{25}{121} a^{3} + \frac{43}{121} a^{2} - \frac{54}{121} a + \frac{49}{121}$, $\frac{1}{121} a^{17} + \frac{2}{121} a^{14} - \frac{1}{121} a^{13} - \frac{1}{121} a^{12} + \frac{2}{121} a^{11} - \frac{2}{121} a^{10} - \frac{5}{121} a^{9} + \frac{4}{121} a^{8} + \frac{21}{121} a^{7} - \frac{49}{121} a^{6} + \frac{57}{121} a^{5} - \frac{6}{121} a^{4} + \frac{2}{121} a^{3} + \frac{53}{121} a^{2} + \frac{38}{121} a + \frac{15}{121}$, $\frac{1}{121} a^{18} - \frac{2}{121} a^{14} + \frac{1}{121} a^{13} + \frac{3}{121} a^{12} - \frac{2}{121} a^{11} + \frac{4}{121} a^{10} - \frac{1}{121} a^{9} + \frac{3}{121} a^{8} - \frac{39}{121} a^{7} + \frac{16}{121} a^{6} - \frac{48}{121} a^{5} + \frac{56}{121} a^{4} + \frac{46}{121} a^{3} + \frac{45}{121} a^{2} - \frac{7}{121} a + \frac{31}{121}$, $\frac{1}{9588403} a^{19} + \frac{540}{871673} a^{18} + \frac{26408}{9588403} a^{17} - \frac{11215}{9588403} a^{16} + \frac{15056}{9588403} a^{15} - \frac{334629}{9588403} a^{14} + \frac{334484}{9588403} a^{13} + \frac{20950}{871673} a^{12} - \frac{294797}{9588403} a^{11} + \frac{119023}{9588403} a^{10} + \frac{268171}{9588403} a^{9} + \frac{419691}{9588403} a^{8} - \frac{2373892}{9588403} a^{7} - \frac{2951396}{9588403} a^{6} + \frac{3262513}{9588403} a^{5} + \frac{27872}{871673} a^{4} + \frac{4262611}{9588403} a^{3} - \frac{2762537}{9588403} a^{2} - \frac{1165451}{9588403} a + \frac{1180642}{9588403}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24199.9496545 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.65808176467.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
307Data not computed
7369Data not computed