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Properties

Label 20.0.32567996285...6976.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 47^{10}$
Root discriminant $23.75$
Ramified primes $2, 3, 47$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, 0, 216513, 0, 341172, 0, 303993, 0, 175689, 0, 69741, 0, 19521, 0, 3753, 0, 468, 0, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 33*x^18 + 468*x^16 + 3753*x^14 + 19521*x^12 + 69741*x^10 + 175689*x^8 + 303993*x^6 + 341172*x^4 + 216513*x^2 + 59049)
 
gp: K = bnfinit(x^20 + 33*x^18 + 468*x^16 + 3753*x^14 + 19521*x^12 + 69741*x^10 + 175689*x^8 + 303993*x^6 + 341172*x^4 + 216513*x^2 + 59049, 1)
 

Normalized defining polynomial

\( x^{20} + 33 x^{18} + 468 x^{16} + 3753 x^{14} + 19521 x^{12} + 69741 x^{10} + 175689 x^{8} + 303993 x^{6} + 341172 x^{4} + 216513 x^{2} + 59049 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3256799628512228606896766976=2^{20}\cdot 3^{10}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.75$
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:  $2, 3, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{10935} a^{14} - \frac{2}{3645} a^{12} + \frac{1}{1215} a^{10} + \frac{1}{405} a^{8} - \frac{1}{135} a^{6} - \frac{1}{45} a^{4} + \frac{2}{15} a^{2} - \frac{1}{5}$, $\frac{1}{10935} a^{15} - \frac{2}{3645} a^{13} + \frac{1}{1215} a^{11} + \frac{1}{405} a^{9} - \frac{1}{135} a^{7} - \frac{1}{45} a^{5} + \frac{2}{15} a^{3} - \frac{1}{5} a$, $\frac{1}{32805} a^{16} + \frac{2}{3645} a^{12} - \frac{2}{1215} a^{10} + \frac{1}{405} a^{8} + \frac{2}{135} a^{6} - \frac{2}{15} a^{2} - \frac{2}{5}$, $\frac{1}{32805} a^{17} + \frac{2}{3645} a^{13} - \frac{2}{1215} a^{11} + \frac{1}{405} a^{9} + \frac{2}{135} a^{7} - \frac{2}{15} a^{3} - \frac{2}{5} a$, $\frac{1}{98415} a^{18} + \frac{2}{3645} a^{12} - \frac{1}{1215} a^{10} + \frac{2}{135} a^{6} - \frac{1}{15} a^{2} + \frac{2}{5}$, $\frac{1}{98415} a^{19} + \frac{2}{3645} a^{13} - \frac{1}{1215} a^{11} + \frac{2}{135} a^{7} - \frac{1}{15} a^{3} + \frac{2}{5} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 85624.8097894 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-141}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{-47})\), 5.1.2209.1 x5, 10.0.229345007.1, 10.0.57068376781824.1 x5, 10.2.1214220782592.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ 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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$