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Properties

Label 20.4.23824343228...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 61^{7}\cdot 397^{4}$
Root discriminant $46.65$
Ramified primes $5, 61, 397$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3721, -9272, 3946, 45122, -155528, 332544, -485421, 484980, -433032, 227014, -112213, 9864, 3033, -7955, 1679, -644, -3, -33, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 13*x^18 - 33*x^17 - 3*x^16 - 644*x^15 + 1679*x^14 - 7955*x^13 + 3033*x^12 + 9864*x^11 - 112213*x^10 + 227014*x^9 - 433032*x^8 + 484980*x^7 - 485421*x^6 + 332544*x^5 - 155528*x^4 + 45122*x^3 + 3946*x^2 - 9272*x + 3721)
 
gp: K = bnfinit(x^20 - 5*x^19 + 13*x^18 - 33*x^17 - 3*x^16 - 644*x^15 + 1679*x^14 - 7955*x^13 + 3033*x^12 + 9864*x^11 - 112213*x^10 + 227014*x^9 - 433032*x^8 + 484980*x^7 - 485421*x^6 + 332544*x^5 - 155528*x^4 + 45122*x^3 + 3946*x^2 - 9272*x + 3721, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 13 x^{18} - 33 x^{17} - 3 x^{16} - 644 x^{15} + 1679 x^{14} - 7955 x^{13} + 3033 x^{12} + 9864 x^{11} - 112213 x^{10} + 227014 x^{9} - 433032 x^{8} + 484980 x^{7} - 485421 x^{6} + 332544 x^{5} - 155528 x^{4} + 45122 x^{3} + 3946 x^{2} - 9272 x + 3721 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2382434322822520356765777587890625=5^{15}\cdot 61^{7}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.65$
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:  $5, 61, 397$
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}{2865041705190792974785544558165485179469493922407504669} a^{19} - \frac{595271633475266307601369839548842106101521802451953189}{2865041705190792974785544558165485179469493922407504669} a^{18} + \frac{1353429683741139682209365628526746143456023387218501515}{2865041705190792974785544558165485179469493922407504669} a^{17} + \frac{889374168233171258407127608378522495057121049833990564}{2865041705190792974785544558165485179469493922407504669} a^{16} + \frac{619378085177355647686540995741912903814979483788954867}{2865041705190792974785544558165485179469493922407504669} a^{15} + \frac{742836320905206822425205506341855583247688243913730355}{2865041705190792974785544558165485179469493922407504669} a^{14} + \frac{1142362474274531621958352532288436188849021455492134022}{2865041705190792974785544558165485179469493922407504669} a^{13} + \frac{272691871294723960311865675206270552593501538115998056}{2865041705190792974785544558165485179469493922407504669} a^{12} - \frac{723906642612367897804384740617509697099782939134972884}{2865041705190792974785544558165485179469493922407504669} a^{11} + \frac{599048834973030026108319292866416009560065395508518873}{2865041705190792974785544558165485179469493922407504669} a^{10} + \frac{25190634005119326216504417976934302129113046539347844}{60958334152995595208203075705648620839776466434202227} a^{9} - \frac{882293525297445451370215138781488315558783165864984603}{2865041705190792974785544558165485179469493922407504669} a^{8} - \frac{299726164903529287689297607214542139827025727535278759}{2865041705190792974785544558165485179469493922407504669} a^{7} + \frac{880595633282002126808209182114147908089178848551054043}{2865041705190792974785544558165485179469493922407504669} a^{6} + \frac{10643961061990751650144591997092425052831043647867699}{39247146646449218832678692577609386020130053731609653} a^{5} - \frac{1288228219006841463455196876175268462760202216140958186}{2865041705190792974785544558165485179469493922407504669} a^{4} - \frac{369108226313366932748418872975360022513604293394234231}{2865041705190792974785544558165485179469493922407504669} a^{3} + \frac{1382954148632200406751398800433853959571077779206094542}{2865041705190792974785544558165485179469493922407504669} a^{2} - \frac{1182268249590596254773783273574491603457592571147906576}{2865041705190792974785544558165485179469493922407504669} a - \frac{7919324724660293801125904724762529698062533088786836}{46967896806406442209599091117466970155237605285368929}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  $11$
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:  \( 20042872.3319 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
61Data not computed
397Data not computed