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Properties

Label 20.20.1284064123...4288.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 17^{5}\cdot 53^{14}$
Root discriminant $56.94$
Ramified primes $2, 17, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:F_5$ (as 20T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -208, 780, 252, -6991, 9335, 13316, -29683, -7565, 37354, -2958, -23708, 5346, 7958, -2392, -1366, 461, 109, -38, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 38*x^18 + 109*x^17 + 461*x^16 - 1366*x^15 - 2392*x^14 + 7958*x^13 + 5346*x^12 - 23708*x^11 - 2958*x^10 + 37354*x^9 - 7565*x^8 - 29683*x^7 + 13316*x^6 + 9335*x^5 - 6991*x^4 + 252*x^3 + 780*x^2 - 208*x + 16)
 
gp: K = bnfinit(x^20 - 3*x^19 - 38*x^18 + 109*x^17 + 461*x^16 - 1366*x^15 - 2392*x^14 + 7958*x^13 + 5346*x^12 - 23708*x^11 - 2958*x^10 + 37354*x^9 - 7565*x^8 - 29683*x^7 + 13316*x^6 + 9335*x^5 - 6991*x^4 + 252*x^3 + 780*x^2 - 208*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 38 x^{18} + 109 x^{17} + 461 x^{16} - 1366 x^{15} - 2392 x^{14} + 7958 x^{13} + 5346 x^{12} - 23708 x^{11} - 2958 x^{10} + 37354 x^{9} - 7565 x^{8} - 29683 x^{7} + 13316 x^{6} + 9335 x^{5} - 6991 x^{4} + 252 x^{3} + 780 x^{2} - 208 x + 16 \)

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:  \(128406412316224387779791991644684288=2^{16}\cdot 17^{5}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.94$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{6056} a^{18} - \frac{207}{6056} a^{17} - \frac{67}{6056} a^{16} + \frac{215}{6056} a^{15} + \frac{273}{3028} a^{14} + \frac{151}{3028} a^{13} + \frac{313}{3028} a^{12} - \frac{31}{757} a^{11} + \frac{145}{757} a^{10} - \frac{14}{757} a^{9} - \frac{81}{757} a^{8} - \frac{1}{3028} a^{7} - \frac{2283}{6056} a^{6} + \frac{2775}{6056} a^{5} + \frac{1991}{6056} a^{4} + \frac{2777}{6056} a^{3} - \frac{957}{3028} a^{2} - \frac{344}{757} a + \frac{314}{757}$, $\frac{1}{1206175615376} a^{19} + \frac{12438619}{603087807688} a^{18} - \frac{15132809711}{603087807688} a^{17} + \frac{37016717605}{1206175615376} a^{16} + \frac{3486893487}{301543903844} a^{15} - \frac{35822880897}{301543903844} a^{14} - \frac{18735790853}{150771951922} a^{13} - \frac{36939484555}{603087807688} a^{12} + \frac{2670314050}{75385975961} a^{11} + \frac{35789904711}{301543903844} a^{10} + \frac{115848033363}{603087807688} a^{9} + \frac{9550555289}{75385975961} a^{8} - \frac{78929581093}{1206175615376} a^{7} - \frac{91199523103}{301543903844} a^{6} + \frac{298089513057}{603087807688} a^{5} - \frac{503317354005}{1206175615376} a^{4} + \frac{276924781793}{603087807688} a^{3} + \frac{5562543777}{150771951922} a^{2} + \frac{46246981193}{150771951922} a - \frac{12261958718}{75385975961}$

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:  \( 576057394786 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.47753.1, 5.5.2382032.1, 10.10.300726051798272.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$