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Properties

Label 20.0.15432869384...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{16}\cdot 5^{35}\cdot 11^{12}\cdot 41^{12}\cdot 61^{12}$
Root discriminant $32{,}316.35$
Ramified primes $2, 3, 5, 11, 41, 61$
Class number Not computed
Class group Not computed
Galois group $C_5\times F_5$ (as 20T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67777855648151168856132999894099006781776, 0, 0, 0, 0, -2467172500874840049039927482832, 0, 0, 0, 0, 2128761163619493418224, 0, 0, 0, 0, -851737518, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 851737518*x^15 + 2128761163619493418224*x^10 - 2467172500874840049039927482832*x^5 + 67777855648151168856132999894099006781776)
 
gp: K = bnfinit(x^20 - 851737518*x^15 + 2128761163619493418224*x^10 - 2467172500874840049039927482832*x^5 + 67777855648151168856132999894099006781776, 1)
 

Normalized defining polynomial

\( x^{20} - 851737518 x^{15} + 2128761163619493418224 x^{10} - 2467172500874840049039927482832 x^{5} + 67777855648151168856132999894099006781776 \)

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:  \(1543286938436821377977594364411890979294271079928560077574127388000488281250000000000000000=2^{16}\cdot 3^{16}\cdot 5^{35}\cdot 11^{12}\cdot 41^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32{,}316.35$
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:  $2, 3, 5, 11, 41, 61$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3}$, $\frac{1}{27} a^{4}$, $\frac{1}{162} a^{5}$, $\frac{1}{162} a^{6}$, $\frac{1}{486} a^{7}$, $\frac{1}{88938} a^{8} + \frac{4}{183} a^{3}$, $\frac{1}{266814} a^{9} + \frac{4}{549} a^{4}$, $\frac{1}{44041919724} a^{10} - \frac{111563}{135931851} a^{5}$, $\frac{1}{44041919724} a^{11} - \frac{111563}{135931851} a^{6}$, $\frac{1}{8059671309492} a^{12} - \frac{5257639}{49751057466} a^{7} + \frac{6}{61} a^{2}$, $\frac{1}{10904735281742676} a^{13} - \frac{5257639}{67313180751498} a^{8} + \frac{12157}{247599} a^{3}$, $\frac{1}{1995566556558909708} a^{14} - \frac{5257639}{12318312077524134} a^{9} + \frac{204734}{45310617} a^{4}$, $\frac{1}{55388680991904225131214787148433096026043022443192} a^{15} + \frac{97950794750563422708529414009652879}{113968479407210339776162113474142172893092638772} a^{10} - \frac{1276566499652656488878241682140739039}{419211827723917111141825448526196310286} a^{5} - \frac{4836171516991455854423877}{168150118924425854332852889}$, $\frac{1}{55388680991904225131214787148433096026043022443192} a^{16} + \frac{97950794750563422708529414009652879}{113968479407210339776162113474142172893092638772} a^{11} - \frac{1276566499652656488878241682140739039}{419211827723917111141825448526196310286} a^{6} - \frac{4836171516991455854423877}{168150118924425854332852889} a$, $\frac{1}{4571394008304831412754550027721628714317408771303965336} a^{17} - \frac{252254439436463720364744384868946656603}{4703080255457645486372993855680687977692807377884738} a^{12} - \frac{8031528303875592393136970330771486348627}{103796429332614152801604839229637680230503314} a^{7} - \frac{817887014619924346930850875973}{13877933765189639035653347487837} a^{2}$, $\frac{1}{836565103519784148534082655073058054720085805148625656488} a^{18} + \frac{124504764302449352828666461879420457981}{5163982120492494744037547253537395399506702500917442324} a^{13} - \frac{6581705816019735278642184128565644769959}{9497373283934194981346842789511847741091053231} a^{8} + \frac{102064090648814336923207857847121}{2539661879029703943524562590274171} a^{3}$, $\frac{1}{69044227688798345130963443771144700430212841756331521306924104} a^{19} - \frac{11008094117955517281005226396699811976483}{142066312116869022903216962492067284835828892502739755775564} a^{14} - \frac{591338014545553076081610442370449935746034955}{783846709242940914395498975946781329615467896314123} a^{9} + \frac{301651542982923707391880616594460778}{209605913861958555570912724263098155143} a^{4}$

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

Class group and class number

Not computed

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:  \( -\frac{13852570316464705}{227936958814420679552324226948284345786185277544} a^{15} + \frac{6941298393522201182393795}{12663164378578926641795790386015796988121404308} a^{10} - \frac{133906543598983046989762969999}{1257635483171751333425476345578588930858} a^{5} + \frac{148952719988310316536440805}{168150118924425854332852889} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times F_5$ (as 20T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 25 conjugacy class representatives for $C_5\times F_5$
Character table for $C_5\times F_5$ is not computed

Intermediate fields

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

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

Sibling fields

Degree 25 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 R R R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ R $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
5Data not computed
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$41$41.5.4.3$x^{5} - 1476$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.5$x^{5} - 53136$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.0.1$x^{5} - x + 7$$1$$5$$0$$C_5$$[\ ]^{5}$
$61$61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.5$x^{5} - 976$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.3$x^{5} - 244$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$