↑

Properties

Label 20.0.89224910880...7504.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{2}\cdot 1871^{2}$
Root discriminant $39.58$
Ramified primes $2, 11, 23, 1871$
Class number $72$ (GRH)
Class group $[2, 36]$ (GRH)
Galois group 20T341

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![108019, -164600, 380763, -450102, 483947, -315514, 184343, -73634, 30228, -24252, 21065, -6690, -1375, 572, 860, -408, -83, 74, -1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - x^18 + 74*x^17 - 83*x^16 - 408*x^15 + 860*x^14 + 572*x^13 - 1375*x^12 - 6690*x^11 + 21065*x^10 - 24252*x^9 + 30228*x^8 - 73634*x^7 + 184343*x^6 - 315514*x^5 + 483947*x^4 - 450102*x^3 + 380763*x^2 - 164600*x + 108019)
 
gp: K = bnfinit(x^20 - 6*x^19 - x^18 + 74*x^17 - 83*x^16 - 408*x^15 + 860*x^14 + 572*x^13 - 1375*x^12 - 6690*x^11 + 21065*x^10 - 24252*x^9 + 30228*x^8 - 73634*x^7 + 184343*x^6 - 315514*x^5 + 483947*x^4 - 450102*x^3 + 380763*x^2 - 164600*x + 108019, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - x^{18} + 74 x^{17} - 83 x^{16} - 408 x^{15} + 860 x^{14} + 572 x^{13} - 1375 x^{12} - 6690 x^{11} + 21065 x^{10} - 24252 x^{9} + 30228 x^{8} - 73634 x^{7} + 184343 x^{6} - 315514 x^{5} + 483947 x^{4} - 450102 x^{3} + 380763 x^{2} - 164600 x + 108019 \)

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:  \(89224910880483379681166705557504=2^{20}\cdot 11^{16}\cdot 23^{2}\cdot 1871^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.58$
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:  $2, 11, 23, 1871$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is 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}{72688886571442833602456302878226167399609160371919185991} a^{19} - \frac{16772915856700940809019802789595701318279165426858638214}{72688886571442833602456302878226167399609160371919185991} a^{18} - \frac{10946076211001416533762762757344968875120023243196848552}{72688886571442833602456302878226167399609160371919185991} a^{17} - \frac{1285757180879337859755713057499708571023210261558988944}{72688886571442833602456302878226167399609160371919185991} a^{16} - \frac{32322883894685817243596625713285852291983373584530285486}{72688886571442833602456302878226167399609160371919185991} a^{15} - \frac{9203756424712026321080423902219342177951987185801388706}{72688886571442833602456302878226167399609160371919185991} a^{14} + \frac{21920329115975632036612821714684293740333237947910072565}{72688886571442833602456302878226167399609160371919185991} a^{13} - \frac{31238462923406508913124893481172442942354038087406613327}{72688886571442833602456302878226167399609160371919185991} a^{12} - \frac{32934949367207942115162529772122554142692152595581582420}{72688886571442833602456302878226167399609160371919185991} a^{11} - \frac{2844142500685286272467683278852160905215411008810939187}{72688886571442833602456302878226167399609160371919185991} a^{10} - \frac{16985261734059710361972577372683148878661173643672829989}{72688886571442833602456302878226167399609160371919185991} a^{9} + \frac{4095458747590444334157442465850318148080770380983250239}{72688886571442833602456302878226167399609160371919185991} a^{8} - \frac{26665354715206485355903884376442450265967462516935919684}{72688886571442833602456302878226167399609160371919185991} a^{7} - \frac{30314038797083463215517455434466514299736805061215169848}{72688886571442833602456302878226167399609160371919185991} a^{6} - \frac{6148040647894890589792121055207067720208758318326759621}{72688886571442833602456302878226167399609160371919185991} a^{5} + \frac{9750697293864505280701716999295306126098854890578681361}{72688886571442833602456302878226167399609160371919185991} a^{4} - \frac{8466530155340304449055927428520240288337650790865861504}{72688886571442833602456302878226167399609160371919185991} a^{3} - \frac{3851234606791293115882486824193588837975134748121405836}{72688886571442833602456302878226167399609160371919185991} a^{2} - \frac{16235532521677475051141041498773922140882255274736248388}{72688886571442833602456302878226167399609160371919185991} a - \frac{2321959945482446046475151324530211703592407077509216364}{72688886571442833602456302878226167399609160371919185991}$

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

Class group and class number

$C_{2}\times C_{36}$, which has order $72$ (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:  \( 447363.730538 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T341:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 224 conjugacy class representatives for t20n341 are not computed
Character table for t20n341 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.401065466351.1, 10.0.9445893863498752.1, 10.10.5048580365312.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.9$x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.10.9$x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
1871Data not computed