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Properties

Label 20.20.6119366577...0625.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{4}\cdot 5^{14}\cdot 23^{4}\cdot 89^{7}$
Root discriminant $34.62$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -76, -725, 1475, 10432, -12012, -31327, 24983, 43333, -21581, -32275, 8956, 13516, -1894, -3262, 197, 450, -8, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 33*x^18 - 8*x^17 + 450*x^16 + 197*x^15 - 3262*x^14 - 1894*x^13 + 13516*x^12 + 8956*x^11 - 32275*x^10 - 21581*x^9 + 43333*x^8 + 24983*x^7 - 31327*x^6 - 12012*x^5 + 10432*x^4 + 1475*x^3 - 725*x^2 - 76*x + 1)
 
gp: K = bnfinit(x^20 - 33*x^18 - 8*x^17 + 450*x^16 + 197*x^15 - 3262*x^14 - 1894*x^13 + 13516*x^12 + 8956*x^11 - 32275*x^10 - 21581*x^9 + 43333*x^8 + 24983*x^7 - 31327*x^6 - 12012*x^5 + 10432*x^4 + 1475*x^3 - 725*x^2 - 76*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 33 x^{18} - 8 x^{17} + 450 x^{16} + 197 x^{15} - 3262 x^{14} - 1894 x^{13} + 13516 x^{12} + 8956 x^{11} - 32275 x^{10} - 21581 x^{9} + 43333 x^{8} + 24983 x^{7} - 31327 x^{6} - 12012 x^{5} + 10432 x^{4} + 1475 x^{3} - 725 x^{2} - 76 x + 1 \)

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:  \(6119366577566395275933837890625=3^{4}\cdot 5^{14}\cdot 23^{4}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.62$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{780090402954419396011} a^{19} + \frac{129796252612246045066}{780090402954419396011} a^{18} + \frac{120301632372858482179}{780090402954419396011} a^{17} - \frac{290394333059837103003}{780090402954419396011} a^{16} + \frac{330497207519678975852}{780090402954419396011} a^{15} + \frac{221308453146308106051}{780090402954419396011} a^{14} + \frac{357784374894195285226}{780090402954419396011} a^{13} + \frac{81009202680335890753}{780090402954419396011} a^{12} + \frac{265801181159797012669}{780090402954419396011} a^{11} + \frac{196185285983195769484}{780090402954419396011} a^{10} + \frac{19608227032930995361}{780090402954419396011} a^{9} + \frac{334039407712711806284}{780090402954419396011} a^{8} + \frac{150225492118547291823}{780090402954419396011} a^{7} + \frac{186580797178763309042}{780090402954419396011} a^{6} - \frac{26054289580510310115}{780090402954419396011} a^{5} - \frac{34167665614797528475}{780090402954419396011} a^{4} - \frac{244270239217526470484}{780090402954419396011} a^{3} + \frac{186686888361409919905}{780090402954419396011} a^{2} - \frac{39715450006129111769}{780090402954419396011} a - \frac{114729128560451221974}{780090402954419396011}$

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:  \( 729718318.944 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 5.5.767625.1, 10.10.2946240703125.1

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 24 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.4.2.1$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}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
89.6.3.2$x^{6} - 7921 x^{2} + 4934783$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$