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Properties

Label 20.20.6513733425...1536.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{18}\cdot 1429^{9}$
Root discriminant $49.06$
Ramified primes $2, 1429$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:D_5$ (as 20T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -128, 260, 3688, -2245, -25596, 18102, 62344, -63708, -41248, 65496, 1368, -27246, 6348, 4826, -1984, -279, 220, -10, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 10*x^18 + 220*x^17 - 279*x^16 - 1984*x^15 + 4826*x^14 + 6348*x^13 - 27246*x^12 + 1368*x^11 + 65496*x^10 - 41248*x^9 - 63708*x^8 + 62344*x^7 + 18102*x^6 - 25596*x^5 - 2245*x^4 + 3688*x^3 + 260*x^2 - 128*x - 4)
 
gp: K = bnfinit(x^20 - 8*x^19 - 10*x^18 + 220*x^17 - 279*x^16 - 1984*x^15 + 4826*x^14 + 6348*x^13 - 27246*x^12 + 1368*x^11 + 65496*x^10 - 41248*x^9 - 63708*x^8 + 62344*x^7 + 18102*x^6 - 25596*x^5 - 2245*x^4 + 3688*x^3 + 260*x^2 - 128*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 10 x^{18} + 220 x^{17} - 279 x^{16} - 1984 x^{15} + 4826 x^{14} + 6348 x^{13} - 27246 x^{12} + 1368 x^{11} + 65496 x^{10} - 41248 x^{9} - 63708 x^{8} + 62344 x^{7} + 18102 x^{6} - 25596 x^{5} - 2245 x^{4} + 3688 x^{3} + 260 x^{2} - 128 x - 4 \)

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:  \(6513733425152557402011290689601536=2^{18}\cdot 1429^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.06$
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:  $2, 1429$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} + \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} + \frac{1}{8} a^{7} + \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{18} + \frac{1}{8} a^{13} + \frac{1}{8} a^{12} + \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{29912356217768127752} a^{19} - \frac{1449439920111015667}{29912356217768127752} a^{18} - \frac{119126569549285877}{3739044527221015969} a^{17} + \frac{491066292144901485}{14956178108884063876} a^{16} - \frac{1609382866637078915}{29912356217768127752} a^{15} - \frac{924953050052920883}{14956178108884063876} a^{14} + \frac{2404049053197224527}{29912356217768127752} a^{13} + \frac{287346259149027421}{2719305110706193432} a^{12} - \frac{1840734755445830045}{29912356217768127752} a^{11} - \frac{6386500619791795303}{29912356217768127752} a^{10} + \frac{1436945791654803693}{14956178108884063876} a^{9} + \frac{906914580836402859}{29912356217768127752} a^{8} - \frac{6627808356714862255}{14956178108884063876} a^{7} - \frac{13903585848668520531}{29912356217768127752} a^{6} + \frac{7206580440730369847}{29912356217768127752} a^{5} + \frac{464129892467460451}{14956178108884063876} a^{4} + \frac{1187681504137656347}{29912356217768127752} a^{3} - \frac{375851687413416387}{3739044527221015969} a^{2} + \frac{4202256778225413079}{14956178108884063876} a - \frac{728835350919321968}{3739044527221015969}$

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:  \( 117630359267 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.5.2042041.1, 10.10.266875612523584.1

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
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
1429Data not computed