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Properties

Label 20.20.8230707706...8125.2
Degree $20$
Signature $[20, 0]$
Discriminant $5^{13}\cdot 19^{10}\cdot 43^{8}\cdot 97^{2}$
Root discriminant $88.26$
Ramified primes $5, 19, 43, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3781, 43985, -87875, -335464, 477805, 821513, -901175, -963875, 771844, 594912, -354586, -201486, 95351, 38099, -15219, -3916, 1369, 200, -61, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 61*x^18 + 200*x^17 + 1369*x^16 - 3916*x^15 - 15219*x^14 + 38099*x^13 + 95351*x^12 - 201486*x^11 - 354586*x^10 + 594912*x^9 + 771844*x^8 - 963875*x^7 - 901175*x^6 + 821513*x^5 + 477805*x^4 - 335464*x^3 - 87875*x^2 + 43985*x + 3781)
 
gp: K = bnfinit(x^20 - 4*x^19 - 61*x^18 + 200*x^17 + 1369*x^16 - 3916*x^15 - 15219*x^14 + 38099*x^13 + 95351*x^12 - 201486*x^11 - 354586*x^10 + 594912*x^9 + 771844*x^8 - 963875*x^7 - 901175*x^6 + 821513*x^5 + 477805*x^4 - 335464*x^3 - 87875*x^2 + 43985*x + 3781, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 61 x^{18} + 200 x^{17} + 1369 x^{16} - 3916 x^{15} - 15219 x^{14} + 38099 x^{13} + 95351 x^{12} - 201486 x^{11} - 354586 x^{10} + 594912 x^{9} + 771844 x^{8} - 963875 x^{7} - 901175 x^{6} + 821513 x^{5} + 477805 x^{4} - 335464 x^{3} - 87875 x^{2} + 43985 x + 3781 \)

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:  \(823070770673063523939149301768798828125=5^{13}\cdot 19^{10}\cdot 43^{8}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.26$
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:  $5, 19, 43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{779} a^{18} - \frac{8}{779} a^{17} + \frac{90}{779} a^{16} - \frac{333}{779} a^{15} + \frac{168}{779} a^{14} + \frac{188}{779} a^{13} + \frac{126}{779} a^{12} - \frac{16}{779} a^{11} - \frac{11}{41} a^{10} - \frac{14}{779} a^{9} - \frac{28}{779} a^{8} - \frac{240}{779} a^{7} - \frac{180}{779} a^{6} - \frac{48}{779} a^{5} - \frac{67}{779} a^{4} - \frac{17}{41} a^{3} - \frac{9}{41} a^{2} - \frac{4}{41} a + \frac{19}{41}$, $\frac{1}{415636433433390651918771526014722578981} a^{19} - \frac{30902491215509672451286710026676441}{59376633347627235988395932287817511283} a^{18} + \frac{79766427446354002895254491006966069}{6203528857214785849533903373354068343} a^{17} - \frac{134708227664162345807834553448448327703}{415636433433390651918771526014722578981} a^{16} - \frac{109024648210212568427243683173600490748}{415636433433390651918771526014722578981} a^{15} + \frac{177336059985915114744875922114686067257}{415636433433390651918771526014722578981} a^{14} + \frac{26356481856499175219208203124292589370}{59376633347627235988395932287817511283} a^{13} + \frac{308191523392051266226590405640203534}{21875601759652139574672185579722240999} a^{12} - \frac{5088948072522555761668274541931130843}{415636433433390651918771526014722578981} a^{11} - \frac{19156721695750152663239145587138500179}{59376633347627235988395932287817511283} a^{10} - \frac{122925277654921961755849221595608993373}{415636433433390651918771526014722578981} a^{9} + \frac{76246850278453786386495535281676128203}{415636433433390651918771526014722578981} a^{8} + \frac{181275943747056932914760916225802576444}{415636433433390651918771526014722578981} a^{7} - \frac{7124373660345695869305347925130040284}{59376633347627235988395932287817511283} a^{6} - \frac{201536261009438413951581628683881685242}{415636433433390651918771526014722578981} a^{5} - \frac{148020357013483421008631463178089237765}{415636433433390651918771526014722578981} a^{4} - \frac{7512721337469745569658764663050466130}{21875601759652139574672185579722240999} a^{3} + \frac{4678609354133821462125751731535172206}{21875601759652139574672185579722240999} a^{2} + \frac{1697534892407304063784234760445680031}{21875601759652139574672185579722240999} a + \frac{717359830280790304863707621169180314}{21875601759652139574672185579722240999}$

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:  \( 15171524862900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.667489.1, 10.10.1392317391003125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ $20$ R $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$