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Properties

Label 20.4.86615880553...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 47^{8}\cdot 193^{2}$
Root discriminant $17.66$
Ramified primes $5, 47, 193$
Class number $1$ (GRH)
Class group Trivial (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, 0, 3, -4, -20, -23, 8, 76, 57, -33, -97, -52, -67, 15, 34, -29, 3, 11, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 4*x^18 + 11*x^17 + 3*x^16 - 29*x^15 + 34*x^14 + 15*x^13 - 67*x^12 - 52*x^11 - 97*x^10 - 33*x^9 + 57*x^8 + 76*x^7 + 8*x^6 - 23*x^5 - 20*x^4 - 4*x^3 + 3*x^2 + 1)
 
gp: K = bnfinit(x^20 - x^19 - 4*x^18 + 11*x^17 + 3*x^16 - 29*x^15 + 34*x^14 + 15*x^13 - 67*x^12 - 52*x^11 - 97*x^10 - 33*x^9 + 57*x^8 + 76*x^7 + 8*x^6 - 23*x^5 - 20*x^4 - 4*x^3 + 3*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 4 x^{18} + 11 x^{17} + 3 x^{16} - 29 x^{15} + 34 x^{14} + 15 x^{13} - 67 x^{12} - 52 x^{11} - 97 x^{10} - 33 x^{9} + 57 x^{8} + 76 x^{7} + 8 x^{6} - 23 x^{5} - 20 x^{4} - 4 x^{3} + 3 x^{2} + 1 \)

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:  \(8661588055311870009765625=5^{10}\cdot 47^{8}\cdot 193^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.66$
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:  $5, 47, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{5}{11} a^{14} - \frac{3}{11} a^{13} + \frac{3}{11} a^{11} + \frac{1}{11} a^{10} + \frac{2}{11} a^{9} - \frac{3}{11} a^{8} - \frac{4}{11} a^{7} - \frac{2}{11} a^{6} - \frac{5}{11} a^{5} + \frac{3}{11} a^{4} - \frac{1}{11} a^{3} - \frac{1}{11} a^{2} + \frac{3}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{15} - \frac{3}{11} a^{14} + \frac{3}{11} a^{12} + \frac{1}{11} a^{11} + \frac{2}{11} a^{10} - \frac{3}{11} a^{9} - \frac{4}{11} a^{8} - \frac{2}{11} a^{7} - \frac{5}{11} a^{6} + \frac{3}{11} a^{5} - \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{3}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{11} a^{18} - \frac{3}{11} a^{15} - \frac{3}{11} a^{14} - \frac{4}{11} a^{13} + \frac{1}{11} a^{12} - \frac{2}{11} a^{11} + \frac{3}{11} a^{10} - \frac{3}{11} a^{9} + \frac{2}{11} a^{8} + \frac{4}{11} a^{7} + \frac{2}{11} a^{6} + \frac{2}{11} a^{5} - \frac{5}{11} a^{4} - \frac{3}{11} a^{3} - \frac{5}{11} a^{2} - \frac{4}{11} a - \frac{5}{11}$, $\frac{1}{51101385809966657} a^{19} - \frac{181528396022196}{51101385809966657} a^{18} - \frac{14073059458142}{1087263527871631} a^{17} - \frac{371663547085379}{51101385809966657} a^{16} - \frac{8348569527304364}{51101385809966657} a^{15} - \frac{24147218560321398}{51101385809966657} a^{14} + \frac{17968105604788505}{51101385809966657} a^{13} - \frac{2815470966042041}{51101385809966657} a^{12} - \frac{6390388737446372}{51101385809966657} a^{11} + \frac{18093521206265985}{51101385809966657} a^{10} - \frac{6778689330967244}{51101385809966657} a^{9} - \frac{1612832238444504}{4645580528178787} a^{8} - \frac{17384750204029678}{51101385809966657} a^{7} + \frac{2052637517319576}{4645580528178787} a^{6} - \frac{18698540355655416}{51101385809966657} a^{5} - \frac{462173068978995}{4645580528178787} a^{4} + \frac{10932298465586287}{51101385809966657} a^{3} + \frac{959594342842885}{51101385809966657} a^{2} + \frac{3474183216807071}{51101385809966657} a - \frac{12061477544053336}{51101385809966657}$

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:  $11$
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:  \( 29147.4826388 \) (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

\(\Q(\sqrt{5}) \), 5.1.2209.1, 10.2.941778433.1, 10.2.15249003125.1, 10.2.2943057603125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$193$193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.4.2.1$x^{4} + 1737 x^{2} + 931225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
193.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
193.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$