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Properties

Label 20.12.3197619919...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{13}\cdot 419^{6}\cdot 695771^{2}$
Root discriminant $66.87$
Ramified primes $5, 419, 695771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49341221, -69482714, -68689262, 149491890, -23286812, -80519029, 41873506, 12201676, -13942971, 1374098, 1581758, -436971, -21862, 11122, -611, 1823, -637, 43, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 20*x^18 + 43*x^17 - 637*x^16 + 1823*x^15 - 611*x^14 + 11122*x^13 - 21862*x^12 - 436971*x^11 + 1581758*x^10 + 1374098*x^9 - 13942971*x^8 + 12201676*x^7 + 41873506*x^6 - 80519029*x^5 - 23286812*x^4 + 149491890*x^3 - 68689262*x^2 - 69482714*x + 49341221)
 
gp: K = bnfinit(x^20 - 8*x^19 + 20*x^18 + 43*x^17 - 637*x^16 + 1823*x^15 - 611*x^14 + 11122*x^13 - 21862*x^12 - 436971*x^11 + 1581758*x^10 + 1374098*x^9 - 13942971*x^8 + 12201676*x^7 + 41873506*x^6 - 80519029*x^5 - 23286812*x^4 + 149491890*x^3 - 68689262*x^2 - 69482714*x + 49341221, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 20 x^{18} + 43 x^{17} - 637 x^{16} + 1823 x^{15} - 611 x^{14} + 11122 x^{13} - 21862 x^{12} - 436971 x^{11} + 1581758 x^{10} + 1374098 x^{9} - 13942971 x^{8} + 12201676 x^{7} + 41873506 x^{6} - 80519029 x^{5} - 23286812 x^{4} + 149491890 x^{3} - 68689262 x^{2} - 69482714 x + 49341221 \)

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:  \(3197619919037143226174828028564453125=5^{13}\cdot 419^{6}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.87$
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:  $5, 419, 695771$
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}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{19} - \frac{206679000131835446366985022604861214717270707692984962507907560252526674}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{18} + \frac{1017756908350235353614699894555342218342227897782511091482856528904172389}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{17} + \frac{474265929473174298993951689239975038271331603714565776489746719716259403}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{16} - \frac{391281147818093497739181104688108213289706863211079676592418666140260446}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{15} + \frac{142742136053591189750860496909897203214595626818777817340293935339510497}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{14} - \frac{1062762132672525041719976620897963495504597882757178915959011028402460271}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{13} - \frac{43289099953234666344773456297248691638688164796261408412667648105996098}{201791910754433073613744824411686984222632169033231170888330579240077263} a^{12} + \frac{849421325958458002322423590104585239928771784057804902603045153067251808}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{11} + \frac{685440884808269967076038756139700006895876496268504004342451557230455374}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{10} + \frac{65589148114503103913962626649763030864460255240609515959955242499056805}{201791910754433073613744824411686984222632169033231170888330579240077263} a^{9} + \frac{73139858079355428443347573858042973542406125214296362314440259059003135}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{8} + \frac{361592816721875441795574580337847584296170414212929026101040732184298424}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{7} - \frac{286502480250482298019230274021559326643973441101808212864960861511117613}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{6} - \frac{156300541378344254804809315097949765146377487199423808158419898277177649}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{5} - \frac{42986196377432675086591932094126481334917974981058272233558193529142280}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{4} + \frac{339276881797720827307732507066416574737377007553286446097700260606226442}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{3} - \frac{20003481713036414452662004162481269744545945716793641843969630260339877}{2623294839807629956978682717351930794894218197432005221548297530121004419} a^{2} + \frac{878427652862938260039393574579510042069079848952723059363236006650408049}{2623294839807629956978682717351930794894218197432005221548297530121004419} a + \frac{824777172692325461431896883524864111467615176714528813705363280699286019}{2623294839807629956978682717351930794894218197432005221548297530121004419}$

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:  $15$
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:  \( 114811582105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
419Data not computed
695771Data not computed