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Properties

Label 20.0.64226466175...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 29^{18}$
Root discriminant $69.24$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8705, 43805, 108785, 241895, 299206, -33663, -138042, 66122, 246404, -44433, -34508, 20263, 13368, -3682, -773, 1286, -265, -44, 34, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 34*x^18 - 44*x^17 - 265*x^16 + 1286*x^15 - 773*x^14 - 3682*x^13 + 13368*x^12 + 20263*x^11 - 34508*x^10 - 44433*x^9 + 246404*x^8 + 66122*x^7 - 138042*x^6 - 33663*x^5 + 299206*x^4 + 241895*x^3 + 108785*x^2 + 43805*x + 8705)
 
gp: K = bnfinit(x^20 - 9*x^19 + 34*x^18 - 44*x^17 - 265*x^16 + 1286*x^15 - 773*x^14 - 3682*x^13 + 13368*x^12 + 20263*x^11 - 34508*x^10 - 44433*x^9 + 246404*x^8 + 66122*x^7 - 138042*x^6 - 33663*x^5 + 299206*x^4 + 241895*x^3 + 108785*x^2 + 43805*x + 8705, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 34 x^{18} - 44 x^{17} - 265 x^{16} + 1286 x^{15} - 773 x^{14} - 3682 x^{13} + 13368 x^{12} + 20263 x^{11} - 34508 x^{10} - 44433 x^{9} + 246404 x^{8} + 66122 x^{7} - 138042 x^{6} - 33663 x^{5} + 299206 x^{4} + 241895 x^{3} + 108785 x^{2} + 43805 x + 8705 \)

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:  \(6422646617589481211251988555908203125=5^{15}\cdot 29^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.24$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{20} a^{16} - \frac{3}{20} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{20} a^{12} + \frac{1}{20} a^{10} + \frac{1}{4} a^{9} - \frac{1}{5} a^{8} - \frac{1}{4} a^{6} - \frac{1}{10} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{20} a^{17} - \frac{1}{5} a^{15} - \frac{1}{5} a^{13} + \frac{3}{20} a^{12} + \frac{1}{20} a^{11} - \frac{1}{10} a^{10} + \frac{1}{20} a^{9} + \frac{2}{5} a^{8} + \frac{1}{4} a^{7} + \frac{3}{20} a^{6} + \frac{1}{4} a^{5} + \frac{3}{20} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{200} a^{18} + \frac{1}{50} a^{17} - \frac{1}{40} a^{16} + \frac{27}{200} a^{15} + \frac{41}{200} a^{14} - \frac{19}{100} a^{13} + \frac{1}{100} a^{12} + \frac{1}{100} a^{11} + \frac{3}{50} a^{10} - \frac{63}{200} a^{9} - \frac{19}{200} a^{8} + \frac{53}{200} a^{7} - \frac{39}{100} a^{6} - \frac{3}{40} a^{5} + \frac{61}{200} a^{4} - \frac{1}{40} a^{3} - \frac{1}{20} a^{2} - \frac{2}{5} a + \frac{11}{40}$, $\frac{1}{4417767623503197308841863084163576006521698074676271800} a^{19} - \frac{4704522752173557744083060478540485431738296893689481}{4417767623503197308841863084163576006521698074676271800} a^{18} + \frac{15456649275158044096953985120649012043723564708408867}{883553524700639461768372616832715201304339614935254360} a^{17} - \frac{1294073839300694504149909865756904151271261998302776}{552220952937899663605232885520447000815212259334533975} a^{16} - \frac{459770051367943952924361317222575708598765961945215197}{2208883811751598654420931542081788003260849037338135900} a^{15} + \frac{131206259190456704915243436244819420990711960653122977}{4417767623503197308841863084163576006521698074676271800} a^{14} + \frac{241557850155541824853355717932394553531792212836941603}{1104441905875799327210465771040894001630424518669067950} a^{13} - \frac{91776524444751504883320753718749901133835002966850911}{552220952937899663605232885520447000815212259334533975} a^{12} + \frac{147956440177953395418776980323362624465871959344280161}{2208883811751598654420931542081788003260849037338135900} a^{11} + \frac{377180927098147797965628276770411540150890987957886297}{4417767623503197308841863084163576006521698074676271800} a^{10} - \frac{3233981101365491148902206184905526238385798820683523}{552220952937899663605232885520447000815212259334533975} a^{9} - \frac{257120138516231028819344332953891745181451164850190819}{552220952937899663605232885520447000815212259334533975} a^{8} + \frac{1806960940879583724208783629845713654859687175676481017}{4417767623503197308841863084163576006521698074676271800} a^{7} - \frac{145071835633891436945306190507603486450100456225004909}{883553524700639461768372616832715201304339614935254360} a^{6} - \frac{94459808901066370219068292231294072102712117594726893}{552220952937899663605232885520447000815212259334533975} a^{5} - \frac{188027388157942395766016267045960276088887375336931311}{441776762350319730884186308416357600652169807467627180} a^{4} - \frac{38884064948256819943441123028905661876572153083517337}{883553524700639461768372616832715201304339614935254360} a^{3} - \frac{166197155680780685378773434800219902822174675551943453}{441776762350319730884186308416357600652169807467627180} a^{2} + \frac{285185942889398450066857956079879574971509115006167011}{883553524700639461768372616832715201304339614935254360} a - \frac{65507706058383256574616251776933304950429229333740807}{176710704940127892353674523366543040260867922987050872}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 10016717916.562931 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.105125.2, 5.1.88410125.1, 10.2.39081751012578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$29.10.9.2$x^{10} + 58$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$
29.10.9.2$x^{10} + 58$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$