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Properties

Label 20.8.13186009861...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{38}\cdot 5^{12}\cdot 97\cdot 1193^{4}$
Root discriminant $50.82$
Ramified primes $2, 5, 97, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2209, -11392, 134326, -92876, -336023, 481528, -248298, -27684, 96318, -50860, 2368, 4928, 635, 12, -396, 224, -40, -8, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - 8*x^17 - 40*x^16 + 224*x^15 - 396*x^14 + 12*x^13 + 635*x^12 + 4928*x^11 + 2368*x^10 - 50860*x^9 + 96318*x^8 - 27684*x^7 - 248298*x^6 + 481528*x^5 - 336023*x^4 - 92876*x^3 + 134326*x^2 - 11392*x - 2209)
 
gp: K = bnfinit(x^20 + 2*x^18 - 8*x^17 - 40*x^16 + 224*x^15 - 396*x^14 + 12*x^13 + 635*x^12 + 4928*x^11 + 2368*x^10 - 50860*x^9 + 96318*x^8 - 27684*x^7 - 248298*x^6 + 481528*x^5 - 336023*x^4 - 92876*x^3 + 134326*x^2 - 11392*x - 2209, 1)
 

Normalized defining polynomial

\( x^{20} + 2 x^{18} - 8 x^{17} - 40 x^{16} + 224 x^{15} - 396 x^{14} + 12 x^{13} + 635 x^{12} + 4928 x^{11} + 2368 x^{10} - 50860 x^{9} + 96318 x^{8} - 27684 x^{7} - 248298 x^{6} + 481528 x^{5} - 336023 x^{4} - 92876 x^{3} + 134326 x^{2} - 11392 x - 2209 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13186009861649028087808000000000000=2^{38}\cdot 5^{12}\cdot 97\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.82$
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:  $2, 5, 97, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{28} a^{18} - \frac{1}{14} a^{17} - \frac{9}{28} a^{16} + \frac{3}{7} a^{15} - \frac{13}{28} a^{14} - \frac{1}{2} a^{13} - \frac{5}{28} a^{12} + \frac{2}{7} a^{11} - \frac{3}{14} a^{10} + \frac{1}{7} a^{9} - \frac{1}{2} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{14} a^{4} + \frac{2}{7} a^{3} + \frac{11}{28} a^{2} - \frac{1}{14} a - \frac{11}{28}$, $\frac{1}{78805973847902685608794638238348822338212847442538211164} a^{19} + \frac{139961133219020572826200043070306205329602870797998253}{19701493461975671402198659559587205584553211860634552791} a^{18} - \frac{2641471021440316070051502894019666149776095371945790577}{78805973847902685608794638238348822338212847442538211164} a^{17} - \frac{7977325495504716132898154383493183583875000235787409263}{39402986923951342804397319119174411169106423721269105582} a^{16} - \frac{23085839760860479512991020833691724243926046635876174049}{78805973847902685608794638238348822338212847442538211164} a^{15} - \frac{391371287108275921201742227379182511430643664335148424}{19701493461975671402198659559587205584553211860634552791} a^{14} - \frac{4225227694502950996395275699659483553172071808715235097}{78805973847902685608794638238348822338212847442538211164} a^{13} - \frac{18220714381692346000110649352357891458027088051287334655}{39402986923951342804397319119174411169106423721269105582} a^{12} - \frac{18380660578348766713537840112539938143033230461701269427}{39402986923951342804397319119174411169106423721269105582} a^{11} + \frac{1763938767338543869547084147568223980105831890639280483}{19701493461975671402198659559587205584553211860634552791} a^{10} + \frac{10502513089786058739782084008918503348380318488247802883}{39402986923951342804397319119174411169106423721269105582} a^{9} - \frac{9428179907037732881479712927635790623520685938184850250}{19701493461975671402198659559587205584553211860634552791} a^{8} + \frac{609386545405233278381875708630736094335599916790918971}{2814499065996524486028379937083886512079030265804936113} a^{7} - \frac{144418192912321391488483693896828530276473574539141180}{371726291735390026456578482256362369519871921898765147} a^{6} + \frac{5972459546749520592532569672368148830221102134098627879}{39402986923951342804397319119174411169106423721269105582} a^{5} + \frac{503500279190682798591534142396961703951077214593222296}{2814499065996524486028379937083886512079030265804936113} a^{4} - \frac{32843898792463631872521487828632368406089528107523486813}{78805973847902685608794638238348822338212847442538211164} a^{3} + \frac{5899030113716513866550200764584291701926490723443646169}{19701493461975671402198659559587205584553211860634552791} a^{2} + \frac{32896037363034491816608070930001982814799068020760787697}{78805973847902685608794638238348822338212847442538211164} a + \frac{5889859090065270035295451647651999477425932496054126615}{39402986923951342804397319119174411169106423721269105582}$

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:  $13$
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:  \( 6655168720.15 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
5Data not computed
97Data not computed
1193Data not computed