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Properties

Label 20.12.3220908674...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 19^{5}\cdot 97^{2}\cdot 1699^{5}$
Root discriminant $47.36$
Ramified primes $5, 19, 97, 1699$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-143641, 438442, 346625, -1638254, 439442, 1323427, -755445, -244046, 303594, -80000, -25053, 25590, -6317, -94, 891, -281, -36, 8, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 + 8*x^17 - 36*x^16 - 281*x^15 + 891*x^14 - 94*x^13 - 6317*x^12 + 25590*x^11 - 25053*x^10 - 80000*x^9 + 303594*x^8 - 244046*x^7 - 755445*x^6 + 1323427*x^5 + 439442*x^4 - 1638254*x^3 + 346625*x^2 + 438442*x - 143641)
 
gp: K = bnfinit(x^20 - x^19 - x^18 + 8*x^17 - 36*x^16 - 281*x^15 + 891*x^14 - 94*x^13 - 6317*x^12 + 25590*x^11 - 25053*x^10 - 80000*x^9 + 303594*x^8 - 244046*x^7 - 755445*x^6 + 1323427*x^5 + 439442*x^4 - 1638254*x^3 + 346625*x^2 + 438442*x - 143641, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} + 8 x^{17} - 36 x^{16} - 281 x^{15} + 891 x^{14} - 94 x^{13} - 6317 x^{12} + 25590 x^{11} - 25053 x^{10} - 80000 x^{9} + 303594 x^{8} - 244046 x^{7} - 755445 x^{6} + 1323427 x^{5} + 439442 x^{4} - 1638254 x^{3} + 346625 x^{2} + 438442 x - 143641 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3220908674307440977286699306640625=5^{10}\cdot 19^{5}\cdot 97^{2}\cdot 1699^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.36$
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:  $5, 19, 97, 1699$
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}{735545766793222809888279420914016176199869304414962724377} a^{19} + \frac{361991918970488747478768152869823141119280676639956519946}{735545766793222809888279420914016176199869304414962724377} a^{18} + \frac{197092250961754586275185117661766718419170954859684808692}{735545766793222809888279420914016176199869304414962724377} a^{17} - \frac{261354635864312773239165434887717523389424744716227988695}{735545766793222809888279420914016176199869304414962724377} a^{16} - \frac{61830906930247410509842182054930758164773080073006207660}{735545766793222809888279420914016176199869304414962724377} a^{15} + \frac{326611268586728762296711235850386961804883564282591471674}{735545766793222809888279420914016176199869304414962724377} a^{14} - \frac{17236211484659328373864890762949109663600666448862432456}{43267398046660165287545848289059775070580547318527219081} a^{13} - \frac{348403802504648348483381896153125952303755028324149259428}{735545766793222809888279420914016176199869304414962724377} a^{12} + \frac{17626368246350948055268905115811069678699866367068188506}{38712935094380147888856811627053482957887858127103301283} a^{11} + \frac{257408279434664814050963435696774486701068592496821450118}{735545766793222809888279420914016176199869304414962724377} a^{10} + \frac{17302848407443892313685692453427655071848338988960249552}{735545766793222809888279420914016176199869304414962724377} a^{9} + \frac{37793183398455182376562242534802906974471511997907949188}{735545766793222809888279420914016176199869304414962724377} a^{8} + \frac{290387569961534399323838663543725636320294567427284256266}{735545766793222809888279420914016176199869304414962724377} a^{7} - \frac{356724773754905257921022855564788834150702845961460435265}{735545766793222809888279420914016176199869304414962724377} a^{6} - \frac{280536875465264450543508295300631823416906005835278061910}{735545766793222809888279420914016176199869304414962724377} a^{5} - \frac{278362300112193767696903461617972797623892994225837153180}{735545766793222809888279420914016176199869304414962724377} a^{4} - \frac{28425978021850841209985409118652190398561944228243517529}{735545766793222809888279420914016176199869304414962724377} a^{3} + \frac{6069626963141663043345925236275225827090392119547586049}{38712935094380147888856811627053482957887858127103301283} a^{2} - \frac{344050787759981286957287484236962216021329277447657456586}{735545766793222809888279420914016176199869304414962724377} a - \frac{196237375743457577073322715227244403817817110713254946282}{735545766793222809888279420914016176199869304414962724377}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 1839406559.04 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$19$$\Q_{19}$$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}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
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}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
1699Data not computed