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Properties

Label 20.8.38355547464...9696.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{40}\cdot 11^{18}\cdot 89^{4}$
Root discriminant $84.96$
Ramified primes $2, 11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T314

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![115971361, 0, 41697568, 0, -43809865, 0, -12218580, 0, 1933338, 0, 710622, 0, 6281, 0, -11440, 0, -869, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 869*x^16 - 11440*x^14 + 6281*x^12 + 710622*x^10 + 1933338*x^8 - 12218580*x^6 - 43809865*x^4 + 41697568*x^2 + 115971361)
 
gp: K = bnfinit(x^20 - 869*x^16 - 11440*x^14 + 6281*x^12 + 710622*x^10 + 1933338*x^8 - 12218580*x^6 - 43809865*x^4 + 41697568*x^2 + 115971361, 1)
 

Normalized defining polynomial

\( x^{20} - 869 x^{16} - 11440 x^{14} + 6281 x^{12} + 710622 x^{10} + 1933338 x^{8} - 12218580 x^{6} - 43809865 x^{4} + 41697568 x^{2} + 115971361 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(383555474642332503321230908091527069696=2^{40}\cdot 11^{18}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.96$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{121} a^{14} + \frac{5}{11} a^{8} - \frac{1}{11} a^{6} - \frac{1}{11} a^{4}$, $\frac{1}{121} a^{15} + \frac{5}{11} a^{9} - \frac{1}{11} a^{7} - \frac{1}{11} a^{5}$, $\frac{1}{121} a^{16} - \frac{1}{11} a^{8} - \frac{1}{11} a^{6}$, $\frac{1}{1331} a^{17} - \frac{2}{121} a^{13} + \frac{5}{121} a^{11} - \frac{34}{121} a^{9} - \frac{12}{121} a^{7} - \frac{5}{11} a^{5}$, $\frac{1}{173382532399078067264716306141} a^{18} + \frac{593097652732268300016386}{177101667414788628462427279} a^{16} - \frac{40455773454322871073334212}{15762048399916187933156027831} a^{14} + \frac{78880511920390144757475435}{15762048399916187933156027831} a^{12} + \frac{315047951850990478828754485}{15762048399916187933156027831} a^{10} - \frac{2202659467094562291426421383}{15762048399916187933156027831} a^{8} - \frac{632689440826429297505470380}{1432913490901471630286911621} a^{6} - \frac{88631942127563046918115138}{1432913490901471630286911621} a^{4} + \frac{58090935516452648475412376}{130264862809224693662446511} a^{2} + \frac{513311983140539525716032}{1463650143923872962499399}$, $\frac{1}{173382532399078067264716306141} a^{19} + \frac{669473604359459450182650}{1948118341562674913086700069} a^{17} - \frac{40455773454322871073334212}{15762048399916187933156027831} a^{15} - \frac{311914076507283936229864098}{15762048399916187933156027831} a^{13} + \frac{575577677469439866153647507}{15762048399916187933156027831} a^{11} - \frac{248686524956191886489723718}{15762048399916187933156027831} a^{9} - \frac{706870434247936976762741652}{15762048399916187933156027831} a^{7} - \frac{349161667746012434243008160}{1432913490901471630286911621} a^{5} + \frac{58090935516452648475412376}{130264862809224693662446511} a^{3} + \frac{513311983140539525716032}{1463650143923872962499399} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 245169286905 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T314:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n314
Character table for t20n314 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.19535810978816.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed