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Properties

Label 20.4.20427678222...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 3^{8}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $116.28$
Ramified primes $2, 3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![285605, 0, 11820940, 0, -8095169, 0, -28947445, 0, -18639312, 0, -4926311, 0, -550206, 0, -13869, 0, 1476, 0, 81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 81*x^18 + 1476*x^16 - 13869*x^14 - 550206*x^12 - 4926311*x^10 - 18639312*x^8 - 28947445*x^6 - 8095169*x^4 + 11820940*x^2 + 285605)
 
gp: K = bnfinit(x^20 + 81*x^18 + 1476*x^16 - 13869*x^14 - 550206*x^12 - 4926311*x^10 - 18639312*x^8 - 28947445*x^6 - 8095169*x^4 + 11820940*x^2 + 285605, 1)
 

Normalized defining polynomial

\( x^{20} + 81 x^{18} + 1476 x^{16} - 13869 x^{14} - 550206 x^{12} - 4926311 x^{10} - 18639312 x^{8} - 28947445 x^{6} - 8095169 x^{4} + 11820940 x^{2} + 285605 \)

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:  \(204276782224508435298892398643200000000000=2^{20}\cdot 3^{8}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.28$
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:  $2, 3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{15} a^{16} + \frac{7}{15} a^{14} - \frac{2}{5} a^{12} + \frac{2}{15} a^{10} + \frac{1}{3} a^{8} + \frac{1}{15} a^{6} - \frac{1}{15} a^{4} + \frac{1}{3}$, $\frac{1}{15} a^{17} + \frac{7}{15} a^{15} - \frac{2}{5} a^{13} + \frac{2}{15} a^{11} + \frac{1}{3} a^{9} + \frac{1}{15} a^{7} - \frac{1}{15} a^{5} + \frac{1}{3} a$, $\frac{1}{64269128173942202609889523102441259325} a^{18} + \frac{683945510445064095750775117028346413}{21423042724647400869963174367480419775} a^{16} - \frac{7580447676759100742420309962936464817}{64269128173942202609889523102441259325} a^{14} - \frac{6037526778695335538312395005428834318}{12853825634788440521977904620488251865} a^{12} + \frac{9893713239119196697078121403868330343}{21423042724647400869963174367480419775} a^{10} + \frac{12306818245719483303022936649668601186}{64269128173942202609889523102441259325} a^{8} - \frac{8242291145293744593014420112062601374}{64269128173942202609889523102441259325} a^{6} - \frac{5392841829872882077682743522083900317}{64269128173942202609889523102441259325} a^{4} + \frac{4104344195254722376122415215044326}{10756339443337607131362263280743307} a^{2} + \frac{5658555036471863540492635611302287}{53781697216688035656811316403716535}$, $\frac{1}{64269128173942202609889523102441259325} a^{19} + \frac{683945510445064095750775117028346413}{21423042724647400869963174367480419775} a^{17} - \frac{7580447676759100742420309962936464817}{64269128173942202609889523102441259325} a^{15} - \frac{6037526778695335538312395005428834318}{12853825634788440521977904620488251865} a^{13} + \frac{9893713239119196697078121403868330343}{21423042724647400869963174367480419775} a^{11} + \frac{12306818245719483303022936649668601186}{64269128173942202609889523102441259325} a^{9} - \frac{8242291145293744593014420112062601374}{64269128173942202609889523102441259325} a^{7} - \frac{5392841829872882077682743522083900317}{64269128173942202609889523102441259325} a^{5} + \frac{4104344195254722376122415215044326}{10756339443337607131362263280743307} a^{3} + \frac{5658555036471863540492635611302287}{53781697216688035656811316403716535} a$

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:  $11$
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:  \( 6733502908630 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
239Data not computed