↑

Properties

Label 20.0.15013659868...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{13}\cdot 97^{2}\cdot 1039^{2}\cdot 1049^{4}$
Root discriminant $36.21$
Ramified primes $5, 97, 1039, 1049$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3079229, 3799228, 2792880, 1595619, 824178, 634779, 452704, 331696, 217288, 113482, 56651, 19115, 5367, 1790, 121, -72, -82, -4, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 4*x^18 - 4*x^17 - 82*x^16 - 72*x^15 + 121*x^14 + 1790*x^13 + 5367*x^12 + 19115*x^11 + 56651*x^10 + 113482*x^9 + 217288*x^8 + 331696*x^7 + 452704*x^6 + 634779*x^5 + 824178*x^4 + 1595619*x^3 + 2792880*x^2 + 3799228*x + 3079229)
 
gp: K = bnfinit(x^20 - 3*x^19 + 4*x^18 - 4*x^17 - 82*x^16 - 72*x^15 + 121*x^14 + 1790*x^13 + 5367*x^12 + 19115*x^11 + 56651*x^10 + 113482*x^9 + 217288*x^8 + 331696*x^7 + 452704*x^6 + 634779*x^5 + 824178*x^4 + 1595619*x^3 + 2792880*x^2 + 3799228*x + 3079229, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 4 x^{18} - 4 x^{17} - 82 x^{16} - 72 x^{15} + 121 x^{14} + 1790 x^{13} + 5367 x^{12} + 19115 x^{11} + 56651 x^{10} + 113482 x^{9} + 217288 x^{8} + 331696 x^{7} + 452704 x^{6} + 634779 x^{5} + 824178 x^{4} + 1595619 x^{3} + 2792880 x^{2} + 3799228 x + 3079229 \)

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:  \(15013659868612228667344970703125=5^{13}\cdot 97^{2}\cdot 1039^{2}\cdot 1049^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.21$
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:  $5, 97, 1039, 1049$
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}{58505094910145859838505224372753240436014581951040006277007213343} a^{19} - \frac{1594074522361698287554332321759034076780341577053742479403098734}{58505094910145859838505224372753240436014581951040006277007213343} a^{18} + \frac{11209891406354483280564826084215343651413127727290516167047897444}{58505094910145859838505224372753240436014581951040006277007213343} a^{17} + \frac{26466785361389237888794294453178845035645043340369779541620416804}{58505094910145859838505224372753240436014581951040006277007213343} a^{16} + \frac{18396516994090535353952812324946460665863451245802190698130808259}{58505094910145859838505224372753240436014581951040006277007213343} a^{15} - \frac{9091977189589760766955770256535455729138660514106219751398895916}{58505094910145859838505224372753240436014581951040006277007213343} a^{14} + \frac{9982770173627303226886665846880583890435680230303886543984343505}{58505094910145859838505224372753240436014581951040006277007213343} a^{13} + \frac{15855161087490825644876273359765861642405682245440652742137342066}{58505094910145859838505224372753240436014581951040006277007213343} a^{12} - \frac{21248943315389332792862611361536783586515498276022274863721463223}{58505094910145859838505224372753240436014581951040006277007213343} a^{11} - \frac{5588349085136989542565465593396538522872185503710185695218003500}{58505094910145859838505224372753240436014581951040006277007213343} a^{10} + \frac{1985359651930390695588875581765517972442700302521604938726715100}{4500391916165066141423478797904095418154967842387692790539016411} a^{9} - \frac{19425054183424279011756437779521855312037659600366394688577311082}{58505094910145859838505224372753240436014581951040006277007213343} a^{8} + \frac{12348401807384684951496862489346733293590423270501120615926859198}{58505094910145859838505224372753240436014581951040006277007213343} a^{7} - \frac{1895231688067343108147238429444643478513178124278435230428018924}{4500391916165066141423478797904095418154967842387692790539016411} a^{6} - \frac{1377700013114357713107113635739958252929047919668251682351706639}{58505094910145859838505224372753240436014581951040006277007213343} a^{5} - \frac{4470743116718844472581653082213974722196368878042208307706578886}{58505094910145859838505224372753240436014581951040006277007213343} a^{4} - \frac{972433890390716459125901352031947806804514225559903456750276643}{58505094910145859838505224372753240436014581951040006277007213343} a^{3} + \frac{14935072162005419053005642744486614680330091380363373860106557995}{58505094910145859838505224372753240436014581951040006277007213343} a^{2} + \frac{13518589877641481912428625085342679814863053824876793760833147711}{58505094910145859838505224372753240436014581951040006277007213343} a - \frac{24293103705827411737005819374292290645744564452973895888829238166}{58505094910145859838505224372753240436014581951040006277007213343}$

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:  $9$
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:  \( 13144391.0374 \) (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

\(\Q(\sqrt{5}) \), 10.4.3405971875.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
1039Data not computed
1049Data not computed