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Properties

Label 20.0.12438117883...8208.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 11^{13}$
Root discriminant $31.97$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5153632, 0, 2811072, 0, 489808, 0, -538208, 0, -56320, 0, -1792, 0, 4684, 0, 72, 0, 90, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 + 90*x^16 + 72*x^14 + 4684*x^12 - 1792*x^10 - 56320*x^8 - 538208*x^6 + 489808*x^4 + 2811072*x^2 + 5153632)
 
gp: K = bnfinit(x^20 + 4*x^18 + 90*x^16 + 72*x^14 + 4684*x^12 - 1792*x^10 - 56320*x^8 - 538208*x^6 + 489808*x^4 + 2811072*x^2 + 5153632, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} + 90 x^{16} + 72 x^{14} + 4684 x^{12} - 1792 x^{10} - 56320 x^{8} - 538208 x^{6} + 489808 x^{4} + 2811072 x^{2} + 5153632 \)

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:  \(1243811788377812389377718878208=2^{55}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.97$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{88} a^{12} + \frac{1}{22} a^{10} + \frac{1}{44} a^{8} - \frac{2}{11} a^{6} + \frac{5}{22} a^{4} - \frac{4}{11} a^{2}$, $\frac{1}{88} a^{13} + \frac{1}{22} a^{11} + \frac{1}{44} a^{9} - \frac{2}{11} a^{7} + \frac{5}{22} a^{5} - \frac{4}{11} a^{3}$, $\frac{1}{968} a^{14} + \frac{1}{242} a^{12} + \frac{45}{484} a^{10} + \frac{9}{121} a^{8} - \frac{39}{242} a^{6} + \frac{18}{121} a^{4} - \frac{2}{11} a^{2}$, $\frac{1}{968} a^{15} + \frac{1}{242} a^{13} + \frac{45}{484} a^{11} + \frac{9}{121} a^{9} - \frac{39}{242} a^{7} + \frac{18}{121} a^{5} - \frac{2}{11} a^{3}$, $\frac{1}{404624} a^{16} + \frac{23}{101156} a^{14} + \frac{9}{5324} a^{12} - \frac{2873}{50578} a^{10} + \frac{2050}{25289} a^{8} - \frac{11741}{50578} a^{6} - \frac{925}{4598} a^{4} + \frac{5}{11} a^{2} + \frac{8}{19}$, $\frac{1}{404624} a^{17} + \frac{23}{101156} a^{15} + \frac{9}{5324} a^{13} - \frac{2873}{50578} a^{11} + \frac{2050}{25289} a^{9} - \frac{11741}{50578} a^{7} - \frac{925}{4598} a^{5} + \frac{5}{11} a^{3} + \frac{8}{19} a$, $\frac{1}{3139769250657171002992} a^{18} + \frac{2291154528297859}{3139769250657171002992} a^{16} + \frac{354070295993815121}{784942312664292750748} a^{14} - \frac{1679198839739226249}{784942312664292750748} a^{12} + \frac{38389802254651339467}{392471156332146375374} a^{10} + \frac{8030774746679985232}{196235578166073187687} a^{8} + \frac{643377714407135217}{35679196030195125034} a^{6} + \frac{31940900853740929}{147434694339649277} a^{4} - \frac{60818151940402528}{147434694339649277} a^{2} - \frac{3749412644207389}{13403154030877207}$, $\frac{1}{3139769250657171002992} a^{19} + \frac{2291154528297859}{3139769250657171002992} a^{17} + \frac{354070295993815121}{784942312664292750748} a^{15} - \frac{1679198839739226249}{784942312664292750748} a^{13} + \frac{38389802254651339467}{392471156332146375374} a^{11} + \frac{8030774746679985232}{196235578166073187687} a^{9} + \frac{643377714407135217}{35679196030195125034} a^{7} + \frac{31940900853740929}{147434694339649277} a^{5} - \frac{60818151940402528}{147434694339649277} a^{3} - \frac{3749412644207389}{13403154030877207} a$

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:  $9$
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:  \( 5307069.668434434 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.22528.2, 10.0.479756288.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.8.3$x^{10} - 11 x^{5} + 847$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$