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Properties

Label 20.20.2950690137...0000.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 7919^{2}$
Root discriminant $74.73$
Ramified primes $2, 5, 11, 7919$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![62710561, 0, -244419935, 0, 334595163, 0, -203748744, 0, 65535196, 0, -12254584, 0, 1400389, 0, -99122, 0, 4236, 0, -100, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 100*x^18 + 4236*x^16 - 99122*x^14 + 1400389*x^12 - 12254584*x^10 + 65535196*x^8 - 203748744*x^6 + 334595163*x^4 - 244419935*x^2 + 62710561)
 
gp: K = bnfinit(x^20 - 100*x^18 + 4236*x^16 - 99122*x^14 + 1400389*x^12 - 12254584*x^10 + 65535196*x^8 - 203748744*x^6 + 334595163*x^4 - 244419935*x^2 + 62710561, 1)
 

Normalized defining polynomial

\( x^{20} - 100 x^{18} + 4236 x^{16} - 99122 x^{14} + 1400389 x^{12} - 12254584 x^{10} + 65535196 x^{8} - 203748744 x^{6} + 334595163 x^{4} - 244419935 x^{2} + 62710561 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(29506901376440972086193367040000000000=2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 7919^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.73$
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:  $2, 5, 11, 7919$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{33} a^{10} + \frac{16}{33} a^{8} + \frac{10}{33} a^{6} + \frac{10}{33} a^{4} + \frac{5}{33} a^{2} + \frac{1}{33}$, $\frac{1}{33} a^{11} + \frac{16}{33} a^{9} + \frac{10}{33} a^{7} + \frac{10}{33} a^{5} + \frac{5}{33} a^{3} + \frac{1}{33} a$, $\frac{1}{33} a^{12} - \frac{5}{11} a^{8} + \frac{5}{11} a^{6} + \frac{10}{33} a^{4} - \frac{13}{33} a^{2} - \frac{16}{33}$, $\frac{1}{33} a^{13} - \frac{5}{11} a^{9} + \frac{5}{11} a^{7} + \frac{10}{33} a^{5} - \frac{13}{33} a^{3} - \frac{16}{33} a$, $\frac{1}{33} a^{14} - \frac{3}{11} a^{8} - \frac{5}{33} a^{6} + \frac{5}{33} a^{4} - \frac{7}{33} a^{2} + \frac{5}{11}$, $\frac{1}{33} a^{15} - \frac{3}{11} a^{9} - \frac{5}{33} a^{7} + \frac{5}{33} a^{5} - \frac{7}{33} a^{3} + \frac{5}{11} a$, $\frac{1}{33} a^{16} + \frac{7}{33} a^{8} - \frac{4}{33} a^{6} - \frac{16}{33} a^{4} - \frac{2}{11} a^{2} + \frac{3}{11}$, $\frac{1}{33} a^{17} + \frac{7}{33} a^{9} - \frac{4}{33} a^{7} - \frac{16}{33} a^{5} - \frac{2}{11} a^{3} + \frac{3}{11} a$, $\frac{1}{3558715730545919203440147} a^{18} - \frac{240281210416712785965}{1186238576848639734480049} a^{16} + \frac{839019193099945417852}{107839870622603612225459} a^{14} + \frac{15987352570610446151297}{3558715730545919203440147} a^{12} + \frac{17326881353972157449089}{1186238576848639734480049} a^{10} - \frac{7534647303747564995038}{323519611867810836676377} a^{8} - \frac{1336561416949185243845741}{3558715730545919203440147} a^{6} + \frac{265955001019191967674931}{3558715730545919203440147} a^{4} + \frac{768450394603807973117501}{3558715730545919203440147} a^{2} + \frac{8815070874260138256}{149796511787932786271}$, $\frac{1}{3558715730545919203440147} a^{19} - \frac{240281210416712785965}{1186238576848639734480049} a^{17} + \frac{839019193099945417852}{107839870622603612225459} a^{15} + \frac{15987352570610446151297}{3558715730545919203440147} a^{13} + \frac{17326881353972157449089}{1186238576848639734480049} a^{11} - \frac{7534647303747564995038}{323519611867810836676377} a^{9} - \frac{1336561416949185243845741}{3558715730545919203440147} a^{7} + \frac{265955001019191967674931}{3558715730545919203440147} a^{5} + \frac{768450394603807973117501}{3558715730545919203440147} a^{3} + \frac{8815070874260138256}{149796511787932786271} a$

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:  $19$
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:  \( 1593960647420 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
7919Data not computed