Properties

Label 20.0.208...625.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.087\times 10^{21}$
Root discriminant $11.64$
Ramified primes $3, 5, 67$
Class number $1$
Class group trivial
Galois group 20T288

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 4, -11, 22, -24, 35, -47, 42, -52, 65, -62, 64, -60, 45, -29, 19, -14, 9, -4, 1]);
 

\( x^{20} - 4 x^{19} + 9 x^{18} - 14 x^{17} + 19 x^{16} - 29 x^{15} + 45 x^{14} - 60 x^{13} + 64 x^{12} - 62 x^{11} + 65 x^{10} - 52 x^{9} + 42 x^{8} - 47 x^{7} + 35 x^{6} - 24 x^{5} + 22 x^{4} - 11 x^{3} + 4 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2086514456522375390625\)\(\medspace = 3^{10}\cdot 5^{8}\cdot 67^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.64$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5, 67$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{21} a^{17} + \frac{1}{7} a^{16} + \frac{3}{7} a^{15} + \frac{10}{21} a^{14} + \frac{1}{3} a^{13} - \frac{3}{7} a^{12} + \frac{5}{21} a^{11} - \frac{4}{21} a^{10} + \frac{2}{21} a^{9} - \frac{5}{21} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{3} + \frac{8}{21} a^{2} + \frac{1}{21} a - \frac{5}{21}$, $\frac{1}{105} a^{18} - \frac{1}{15} a^{16} - \frac{52}{105} a^{15} + \frac{11}{35} a^{14} - \frac{2}{7} a^{13} - \frac{38}{105} a^{12} - \frac{4}{35} a^{11} + \frac{2}{15} a^{10} - \frac{4}{105} a^{9} - \frac{2}{7} a^{8} + \frac{47}{105} a^{7} + \frac{5}{21} a^{6} - \frac{4}{105} a^{5} - \frac{26}{105} a^{4} + \frac{16}{105} a^{3} + \frac{47}{105} a^{2} - \frac{29}{105} a - \frac{34}{105}$, $\frac{1}{207795} a^{19} - \frac{338}{207795} a^{18} + \frac{14}{29685} a^{17} - \frac{7617}{69265} a^{16} + \frac{23498}{69265} a^{15} - \frac{66169}{207795} a^{14} - \frac{86078}{207795} a^{13} - \frac{1626}{69265} a^{12} - \frac{4050}{13853} a^{11} - \frac{55661}{207795} a^{10} - \frac{22391}{69265} a^{9} - \frac{89248}{207795} a^{8} - \frac{103711}{207795} a^{7} - \frac{60359}{207795} a^{6} - \frac{15964}{207795} a^{5} - \frac{14997}{69265} a^{4} - \frac{49006}{207795} a^{3} + \frac{7457}{41559} a^{2} + \frac{18691}{69265} a - \frac{70373}{207795}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{446380}{41559} a^{19} + \frac{507763}{13853} a^{18} - \frac{3138139}{41559} a^{17} + \frac{637280}{5937} a^{16} - \frac{5973073}{41559} a^{15} + \frac{1369838}{5937} a^{14} - \frac{4883015}{13853} a^{13} + \frac{18497107}{41559} a^{12} - \frac{18213646}{41559} a^{11} + \frac{5870395}{13853} a^{10} - \frac{19242005}{41559} a^{9} + \frac{12423695}{41559} a^{8} - \frac{574441}{1979} a^{7} + \frac{4774813}{13853} a^{6} - \frac{2508248}{13853} a^{5} + \frac{6669826}{41559} a^{4} - \frac{6152458}{41559} a^{3} + \frac{477231}{13853} a^{2} - \frac{1074751}{41559} a + \frac{788960}{41559} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 527.926443922 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 527.926443922 \cdot 1}{6\sqrt{2086514456522375390625}}\approx 0.184718396785$

Galois group

20T288:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.5025.1, 10.0.681766875.1, 10.0.1691791875.1, 10.2.45678380625.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
67Data not computed