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Properties

Label 20.6.20562460332...4256.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{48}\cdot 31^{7}\cdot 227^{4}$
Root discriminant $51.96$
Ramified primes $2, 31, 227$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-33664, 64768, -31872, -10624, 88512, -162304, 174368, -51840, -32472, 52944, -1864, -3448, -2936, 2088, -1462, 52, 122, -72, 26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 26*x^18 - 72*x^17 + 122*x^16 + 52*x^15 - 1462*x^14 + 2088*x^13 - 2936*x^12 - 3448*x^11 - 1864*x^10 + 52944*x^9 - 32472*x^8 - 51840*x^7 + 174368*x^6 - 162304*x^5 + 88512*x^4 - 10624*x^3 - 31872*x^2 + 64768*x - 33664)
 
gp: K = bnfinit(x^20 - 4*x^19 + 26*x^18 - 72*x^17 + 122*x^16 + 52*x^15 - 1462*x^14 + 2088*x^13 - 2936*x^12 - 3448*x^11 - 1864*x^10 + 52944*x^9 - 32472*x^8 - 51840*x^7 + 174368*x^6 - 162304*x^5 + 88512*x^4 - 10624*x^3 - 31872*x^2 + 64768*x - 33664, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 26 x^{18} - 72 x^{17} + 122 x^{16} + 52 x^{15} - 1462 x^{14} + 2088 x^{13} - 2936 x^{12} - 3448 x^{11} - 1864 x^{10} + 52944 x^{9} - 32472 x^{8} - 51840 x^{7} + 174368 x^{6} - 162304 x^{5} + 88512 x^{4} - 10624 x^{3} - 31872 x^{2} + 64768 x - 33664 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-20562460332300807986799747065184256=-\,2^{48}\cdot 31^{7}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.96$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{16} - \frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{3}{32} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2}$, $\frac{1}{128} a^{17} - \frac{1}{64} a^{15} + \frac{3}{64} a^{13} - \frac{1}{32} a^{12} - \frac{3}{64} a^{11} + \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{3}{16} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{256} a^{18} + \frac{1}{128} a^{14} + \frac{3}{64} a^{13} - \frac{5}{128} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} - \frac{1}{32} a^{9} - \frac{3}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{32} a^{6} - \frac{3}{8} a^{5} + \frac{5}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{3696904485084629866539739589355241884149915136} a^{19} - \frac{524662792996005381615514543387319576728437}{308075373757052488878311632446270157012492928} a^{18} - \frac{359980108946491011188527233380194052607689}{462113060635578733317467448669405235518739392} a^{17} + \frac{1389646533893320510805342529476688606416369}{924226121271157466634934897338810471037478784} a^{16} - \frac{9167609092403037646477342842785687965616119}{1848452242542314933269869794677620942074957568} a^{15} - \frac{27194710780127492252330068383034284111532069}{924226121271157466634934897338810471037478784} a^{14} + \frac{10364396283723171206442460152547574673049499}{1848452242542314933269869794677620942074957568} a^{13} - \frac{68051190108057818454768400671261031535863}{462113060635578733317467448669405235518739392} a^{12} - \frac{53283702549479114360069261810495706743836321}{924226121271157466634934897338810471037478784} a^{11} + \frac{5186547344288034368948203367917812132049441}{231056530317789366658733724334702617759369696} a^{10} - \frac{8210526198380850388132807229294353908945227}{77018843439263122219577908111567539253123232} a^{9} - \frac{1062554714362894049081223380397804811249741}{19254710859815780554894477027891884813280808} a^{8} + \frac{35708545959339705598519355281156104609730383}{154037686878526244439155816223135078506246464} a^{7} - \frac{2472073497836070609001870383279726381210235}{19254710859815780554894477027891884813280808} a^{6} - \frac{1196969229118001103044447121820993589859953}{2783813618286618875406430413671115876618912} a^{5} + \frac{2425868683208054757655242151947818958867689}{115528265158894683329366862167351308879684848} a^{4} - \frac{417506188097785782011178603801352882309355}{1958106189133808192023167155378835743723472} a^{3} + \frac{378236443995297894321705510547510843615257}{2406838857476972569361809628486485601660101} a^{2} + \frac{3037837877717449600896116418591475920803297}{19254710859815780554894477027891884813280808} a + \frac{10544605253034467067740074510583248956837317}{28882066289723670832341715541837827219921212}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $12$
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:  \( 8804873385.68 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.8.26.3$x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
227Data not computed