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Properties

Label 20.0.17406652163...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 5^{11}\cdot 19^{6}\cdot 461^{4}$
Root discriminant $45.92$
Ramified primes $2, 5, 19, 461$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![369151, -468806, 572299, -863398, 821043, -491660, 219396, -60408, 3245, -2116, 6821, -9576, 9059, -6528, 3584, -1542, 597, -204, 57, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 57*x^18 - 204*x^17 + 597*x^16 - 1542*x^15 + 3584*x^14 - 6528*x^13 + 9059*x^12 - 9576*x^11 + 6821*x^10 - 2116*x^9 + 3245*x^8 - 60408*x^7 + 219396*x^6 - 491660*x^5 + 821043*x^4 - 863398*x^3 + 572299*x^2 - 468806*x + 369151)
 
gp: K = bnfinit(x^20 - 10*x^19 + 57*x^18 - 204*x^17 + 597*x^16 - 1542*x^15 + 3584*x^14 - 6528*x^13 + 9059*x^12 - 9576*x^11 + 6821*x^10 - 2116*x^9 + 3245*x^8 - 60408*x^7 + 219396*x^6 - 491660*x^5 + 821043*x^4 - 863398*x^3 + 572299*x^2 - 468806*x + 369151, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 57 x^{18} - 204 x^{17} + 597 x^{16} - 1542 x^{15} + 3584 x^{14} - 6528 x^{13} + 9059 x^{12} - 9576 x^{11} + 6821 x^{10} - 2116 x^{9} + 3245 x^{8} - 60408 x^{7} + 219396 x^{6} - 491660 x^{5} + 821043 x^{4} - 863398 x^{3} + 572299 x^{2} - 468806 x + 369151 \)

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:  \(1740665216320641860403200000000000=2^{24}\cdot 5^{11}\cdot 19^{6}\cdot 461^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.92$
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:  $2, 5, 19, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{17} + \frac{1}{9} a^{15} - \frac{4}{9} a^{14} + \frac{2}{9} a^{13} + \frac{2}{9} a^{12} + \frac{1}{3} a^{11} + \frac{4}{9} a^{10} - \frac{4}{9} a^{9} + \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{2645890618741791401453839222720854851897376122780746671} a^{19} - \frac{110176034348101518016144485419634846974919525245097586}{2645890618741791401453839222720854851897376122780746671} a^{18} - \frac{350266505047661058624748384572634730012261338190801348}{2645890618741791401453839222720854851897376122780746671} a^{17} - \frac{194217870745914031008013214045762414175858711118045052}{2645890618741791401453839222720854851897376122780746671} a^{16} - \frac{53946651944162482640165185677776397524241037376679715}{881963539580597133817946407573618283965792040926915557} a^{15} + \frac{611556162111093351430453498403746415949791218526553118}{2645890618741791401453839222720854851897376122780746671} a^{14} + \frac{294611177605577463637402673487784145446912589503929238}{2645890618741791401453839222720854851897376122780746671} a^{13} - \frac{728044297650604560444725857681470074095433265022924138}{2645890618741791401453839222720854851897376122780746671} a^{12} + \frac{1085229726591407031576240387015711321643463030881978036}{2645890618741791401453839222720854851897376122780746671} a^{11} - \frac{81341182113665938909968513141855838740963961779655163}{881963539580597133817946407573618283965792040926915557} a^{10} - \frac{215954411316827926993894061881662945074840051576018855}{881963539580597133817946407573618283965792040926915557} a^{9} + \frac{1322749191557938127786108002748413379556844908534794065}{2645890618741791401453839222720854851897376122780746671} a^{8} + \frac{46869110415864071250520412494155312023679082122590516}{293987846526865711272648802524539427988597346975638519} a^{7} - \frac{275302585411977587937051624743729499104575380408003853}{881963539580597133817946407573618283965792040926915557} a^{6} + \frac{361340954554565511827257162518104424610908667627886401}{881963539580597133817946407573618283965792040926915557} a^{5} - \frac{52425595370483626230017834280049698229220893903716110}{2645890618741791401453839222720854851897376122780746671} a^{4} - \frac{78335905886345271957923523078156543138344889898883701}{881963539580597133817946407573618283965792040926915557} a^{3} - \frac{90138442546800437615717929566655426952604316273059066}{293987846526865711272648802524539427988597346975638519} a^{2} + \frac{427382225570504744777548200381891953875265505283968280}{881963539580597133817946407573618283965792040926915557} a - \frac{302378691407306235137957185773237446747413489037195962}{2645890618741791401453839222720854851897376122780746671}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 100730781.481 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.61376064800000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$19.2.1.1$x^{2} - 19$$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}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
461Data not computed