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Properties

Label 20.0.18549495164...5616.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 29^{5}\cdot 53^{14}$
Root discriminant $65.08$
Ramified primes $2, 29, 53$
Class number $820$ (GRH)
Class group $[2, 410]$ (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![141499, 116534, 274509, 251547, 344870, 205103, 186965, 70259, 54982, 10456, 12252, -587, 2547, -658, 212, -28, -1, 28, -8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 8*x^18 + 28*x^17 - x^16 - 28*x^15 + 212*x^14 - 658*x^13 + 2547*x^12 - 587*x^11 + 12252*x^10 + 10456*x^9 + 54982*x^8 + 70259*x^7 + 186965*x^6 + 205103*x^5 + 344870*x^4 + 251547*x^3 + 274509*x^2 + 116534*x + 141499)
 
gp: K = bnfinit(x^20 - 3*x^19 - 8*x^18 + 28*x^17 - x^16 - 28*x^15 + 212*x^14 - 658*x^13 + 2547*x^12 - 587*x^11 + 12252*x^10 + 10456*x^9 + 54982*x^8 + 70259*x^7 + 186965*x^6 + 205103*x^5 + 344870*x^4 + 251547*x^3 + 274509*x^2 + 116534*x + 141499, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 8 x^{18} + 28 x^{17} - x^{16} - 28 x^{15} + 212 x^{14} - 658 x^{13} + 2547 x^{12} - 587 x^{11} + 12252 x^{10} + 10456 x^{9} + 54982 x^{8} + 70259 x^{7} + 186965 x^{6} + 205103 x^{5} + 344870 x^{4} + 251547 x^{3} + 274509 x^{2} + 116534 x + 141499 \)

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:  \(1854949516446736210185316359063535616=2^{16}\cdot 29^{5}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.08$
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:  $2, 29, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{1321390098424255339045680106126647522632225163266038} a^{19} - \frac{91395725871900571112149395336246018378849956020823}{660695049212127669522840053063323761316112581633019} a^{18} + \frac{261508704023936311596336336080035834880142381783971}{1321390098424255339045680106126647522632225163266038} a^{17} - \frac{6685375088495023713430232067635996070223890881537}{38864414659536921736637650180195515371536034213707} a^{16} - \frac{55311363531407676941721197216963830086172537198349}{660695049212127669522840053063323761316112581633019} a^{15} - \frac{115798703989102994038688447369025586333048743139435}{1321390098424255339045680106126647522632225163266038} a^{14} - \frac{311588672388610369121127340716251939845789408954859}{1321390098424255339045680106126647522632225163266038} a^{13} + \frac{203821643166447399421122540877453944541275986325453}{1321390098424255339045680106126647522632225163266038} a^{12} + \frac{20828909772052898047332219880188601547447930434857}{660695049212127669522840053063323761316112581633019} a^{11} - \frac{184495886412757304308432544605118268160603053911478}{660695049212127669522840053063323761316112581633019} a^{10} + \frac{77630445795491866128735780315609987355157962493843}{660695049212127669522840053063323761316112581633019} a^{9} - \frac{477154042234614972785658447509364531935378566950291}{1321390098424255339045680106126647522632225163266038} a^{8} - \frac{230142647575360730548551071560997426816467791102215}{660695049212127669522840053063323761316112581633019} a^{7} - \frac{309930868388343680579790663216476306933094023699957}{660695049212127669522840053063323761316112581633019} a^{6} - \frac{491825596427418533783342558226994936176416260328729}{1321390098424255339045680106126647522632225163266038} a^{5} - \frac{166929745071283754652975594534169370045649441793378}{660695049212127669522840053063323761316112581633019} a^{4} + \frac{116131211417554311289026632887819395284882758950104}{660695049212127669522840053063323761316112581633019} a^{3} - \frac{304581262423010423151179789008690932081998857293790}{660695049212127669522840053063323761316112581633019} a^{2} + \frac{212853268511962165170976208641450301548628665235161}{660695049212127669522840053063323761316112581633019} a + \frac{544007550424520909891147586425910690960524225832903}{1321390098424255339045680106126647522632225163266038}$

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

Class group and class number

$C_{2}\times C_{410}$, which has order $820$ (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:  \( 7826958.70528 \) (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

\(\Q(\sqrt{53}) \), 4.0.81461.1, 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ 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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$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}$