↑

Properties

Label 20.16.3398456149...0000.2
Degree $20$
Signature $[16, 2]$
Discriminant $2^{38}\cdot 5^{14}\cdot 1193^{4}$
Root discriminant $47.49$
Ramified primes $2, 5, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, -3450, 0, -21325, 0, -4500, 0, 67485, 0, -65710, 0, 27365, 0, -5940, 0, 699, 0, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 42*x^18 + 699*x^16 - 5940*x^14 + 27365*x^12 - 65710*x^10 + 67485*x^8 - 4500*x^6 - 21325*x^4 - 3450*x^2 + 25)
 
gp: K = bnfinit(x^20 - 42*x^18 + 699*x^16 - 5940*x^14 + 27365*x^12 - 65710*x^10 + 67485*x^8 - 4500*x^6 - 21325*x^4 - 3450*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 42 x^{18} + 699 x^{16} - 5940 x^{14} + 27365 x^{12} - 65710 x^{10} + 67485 x^{8} - 4500 x^{6} - 21325 x^{4} - 3450 x^{2} + 25 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3398456149909543321600000000000000=2^{38}\cdot 5^{14}\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.49$
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:  $2, 5, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{10} + \frac{1}{5} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{40} a^{13} + \frac{3}{40} a^{11} - \frac{1}{8} a^{10} + \frac{9}{40} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{40} a^{14} - \frac{1}{40} a^{12} - \frac{1}{8} a^{11} - \frac{3}{40} a^{10} + \frac{1}{8} a^{9} - \frac{1}{40} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a$, $\frac{1}{40} a^{15} - \frac{1}{40} a^{12} + \frac{1}{20} a^{10} + \frac{1}{5} a^{9} + \frac{3}{20} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{40} a^{16} - \frac{1}{8} a^{11} + \frac{3}{40} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{40} a^{17} - \frac{1}{40} a^{12} + \frac{3}{40} a^{11} - \frac{3}{40} a^{10} + \frac{1}{8} a^{9} - \frac{9}{40} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{200047469680} a^{18} - \frac{1}{80} a^{17} + \frac{315471123}{100023734840} a^{16} - \frac{1}{80} a^{15} - \frac{86040746}{12502966855} a^{14} - \frac{1}{80} a^{13} + \frac{4255497209}{200047469680} a^{12} + \frac{1}{20} a^{11} + \frac{10134422893}{100023734840} a^{10} + \frac{1}{10} a^{9} - \frac{23883755267}{100023734840} a^{8} - \frac{4201290795}{40009493936} a^{6} + \frac{1}{16} a^{5} - \frac{1193430647}{5001186742} a^{4} - \frac{3}{16} a^{3} - \frac{5377420517}{20004746968} a^{2} + \frac{5}{16} a + \frac{2499005777}{40009493936}$, $\frac{1}{200047469680} a^{19} - \frac{373930225}{40009493936} a^{17} - \frac{1}{80} a^{16} + \frac{224788287}{40009493936} a^{15} - \frac{1}{80} a^{14} + \frac{877451919}{100023734840} a^{13} - \frac{1}{80} a^{12} + \frac{131632139}{5001186742} a^{11} + \frac{1}{20} a^{10} + \frac{465658301}{2500593371} a^{9} + \frac{1}{10} a^{8} - \frac{9202477537}{40009493936} a^{7} - \frac{2045665063}{40009493936} a^{5} - \frac{7}{16} a^{4} - \frac{8254247663}{40009493936} a^{3} + \frac{5}{16} a^{2} + \frac{10001579687}{20004746968} a - \frac{3}{16}$

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:  $17$
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:  \( 15188442027.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
2Data not computed
5Data not computed
1193Data not computed