↑

Properties

Label 20.0.92738759037...5681.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $39.66$
Ramified primes $3, 7, 11$
Class number $330$ (GRH)
Class group $[330]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![278653, -96318, 351000, -158880, 238062, -139074, 123807, -71930, 50255, -27764, 17084, -9042, 4254, -1974, 1108, -452, 132, -52, 27, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 27*x^18 - 52*x^17 + 132*x^16 - 452*x^15 + 1108*x^14 - 1974*x^13 + 4254*x^12 - 9042*x^11 + 17084*x^10 - 27764*x^9 + 50255*x^8 - 71930*x^7 + 123807*x^6 - 139074*x^5 + 238062*x^4 - 158880*x^3 + 351000*x^2 - 96318*x + 278653)
 
gp: K = bnfinit(x^20 - 8*x^19 + 27*x^18 - 52*x^17 + 132*x^16 - 452*x^15 + 1108*x^14 - 1974*x^13 + 4254*x^12 - 9042*x^11 + 17084*x^10 - 27764*x^9 + 50255*x^8 - 71930*x^7 + 123807*x^6 - 139074*x^5 + 238062*x^4 - 158880*x^3 + 351000*x^2 - 96318*x + 278653, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 27 x^{18} - 52 x^{17} + 132 x^{16} - 452 x^{15} + 1108 x^{14} - 1974 x^{13} + 4254 x^{12} - 9042 x^{11} + 17084 x^{10} - 27764 x^{9} + 50255 x^{8} - 71930 x^{7} + 123807 x^{6} - 139074 x^{5} + 238062 x^{4} - 158880 x^{3} + 351000 x^{2} - 96318 x + 278653 \)

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:  \(92738759037689478010716606945681=3^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.66$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(197,·)$, $\chi_{231}(134,·)$, $\chi_{231}(8,·)$, $\chi_{231}(202,·)$, $\chi_{231}(62,·)$, $\chi_{231}(83,·)$, $\chi_{231}(148,·)$, $\chi_{231}(29,·)$, $\chi_{231}(223,·)$, $\chi_{231}(97,·)$, $\chi_{231}(34,·)$, $\chi_{231}(230,·)$, $\chi_{231}(167,·)$, $\chi_{231}(41,·)$, $\chi_{231}(50,·)$, $\chi_{231}(181,·)$, $\chi_{231}(169,·)$, $\chi_{231}(190,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{67108807037207236545887877366240513556890506} a^{19} - \frac{11520241337880660379127872001684666069917615}{67108807037207236545887877366240513556890506} a^{18} - \frac{2218723088015768938527342499327391365388871}{33554403518603618272943938683120256778445253} a^{17} + \frac{10576462886271910849288732076479855525424147}{67108807037207236545887877366240513556890506} a^{16} - \frac{10668277309886523326026289338083169676065205}{67108807037207236545887877366240513556890506} a^{15} + \frac{2922813306093270109541476267802577842323830}{33554403518603618272943938683120256778445253} a^{14} + \frac{5334816130175132081645427359012129501138322}{33554403518603618272943938683120256778445253} a^{13} - \frac{7739187645107550898638533661809248501675950}{33554403518603618272943938683120256778445253} a^{12} - \frac{1106780996430215902802061761442784386340762}{33554403518603618272943938683120256778445253} a^{11} + \frac{5690467644836405312446472135498067257961463}{67108807037207236545887877366240513556890506} a^{10} + \frac{3747212802663590841079736133068552145162806}{33554403518603618272943938683120256778445253} a^{9} - \frac{173104125919454153435192706307075668960318}{500811992814979377208118487807765026543959} a^{8} + \frac{25130227771703158059035993314978066098409211}{67108807037207236545887877366240513556890506} a^{7} - \frac{110884594556252844068625202273428326422193}{500811992814979377208118487807765026543959} a^{6} - \frac{1097522100103608201114003767457389573380521}{33554403518603618272943938683120256778445253} a^{5} + \frac{3714826438560749532468253528585422751058099}{33554403518603618272943938683120256778445253} a^{4} - \frac{7601822259533721462934526471822101575325977}{67108807037207236545887877366240513556890506} a^{3} - \frac{14425614338460686474240251398350068670425330}{33554403518603618272943938683120256778445253} a^{2} - \frac{6944947810120940960406617645800123743864999}{33554403518603618272943938683120256778445253} a - \frac{7787813137125807011922320483848723743265}{500811992814979377208118487807765026543959}$

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

Class group and class number

$C_{330}$, which has order $330$ (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:  \( 125582.779517 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{-7}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\), 10.0.3602729712967.1, 10.0.9630096522760791.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
7Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$