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Properties

Label 20.0.55151002218...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{35}\cdot 11^{12}\cdot 461^{12}$
Root discriminant $4864.94$
Ramified primes $2, 5, 11, 461$
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![526728719890528781692016896, 0, 0, 0, 0, -54627959472340369984, 0, 0, 0, 0, -16449885784464, 0, 0, 0, 0, -2380244, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2380244*x^15 - 16449885784464*x^10 - 54627959472340369984*x^5 + 526728719890528781692016896)
 
gp: K = bnfinit(x^20 - 2380244*x^15 - 16449885784464*x^10 - 54627959472340369984*x^5 + 526728719890528781692016896, 1)
 

Normalized defining polynomial

\( x^{20} - 2380244 x^{15} - 16449885784464 x^{10} - 54627959472340369984 x^{5} + 526728719890528781692016896 \)

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:  \(55151002218183282893941830832304805064658020210266113281250000000000000000=2^{16}\cdot 5^{35}\cdot 11^{12}\cdot 461^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4864.94$
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:  $2, 5, 11, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{8} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{144876522736} a^{10} - \frac{132009165}{1906270036} a^{5} + \frac{740493}{1785601}$, $\frac{1}{144876522736} a^{11} - \frac{132009165}{1906270036} a^{6} + \frac{740493}{1785601} a$, $\frac{1}{1593641750096} a^{12} - \frac{132009165}{20968970396} a^{7} - \frac{2830709}{19641611} a^{2}$, $\frac{1}{1469337693588512} a^{13} + \frac{775719895487}{19333390705112} a^{8} - \frac{1621848262}{9054782671} a^{3}$, $\frac{1}{16162714629473632} a^{14} + \frac{1003217676238}{26583412219529} a^{9} - \frac{30408044537}{199205218762} a^{4}$, $\frac{1}{16837198850459904248741693504} a^{15} + \frac{826376688376766}{263081232038436003886588961} a^{10} - \frac{75021364690897920180}{51879556702511536952591} a^{5} - \frac{6638278225574}{183407187124141}$, $\frac{1}{16837198850459904248741693504} a^{16} + \frac{826376688376766}{263081232038436003886588961} a^{11} - \frac{75021364690897920180}{51879556702511536952591} a^{6} - \frac{6638278225574}{183407187124141} a$, $\frac{1}{85381435370682174445369127758784} a^{17} + \frac{2766778535108198273}{10672679421335271805671140969848} a^{12} - \frac{1157841988655905132916821}{526162464076872007773177922} a^{7} + \frac{379975171395842909}{930057845906519011} a^{2}$, $\frac{1}{1878391578155007837798120810693248} a^{18} + \frac{32359901717730055}{117399473634687989862382550668328} a^{13} + \frac{282002824580090796929155161}{5787787104845592085504957142} a^{8} + \frac{4686901451899384251}{20461272609943418242} a^{3}$, $\frac{1}{9525323692824044745474270631025460608} a^{19} - \frac{7860519008290402305}{2381330923206011186368567657756365152} a^{14} - \frac{3413737757752726776573880688143}{117399473634687989862382550668328} a^{9} + \frac{7756939441467951393849}{51879556702511536952591} a^{4}$

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:  \( -\frac{66778659}{16837198850459904248741693504} a^{15} - \frac{46961849360605}{1052324928153744015546355844} a^{10} - \frac{3849310321399365}{103759113405023073905182} a^{5} + \frac{182430536391150}{183407187124141} \) (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 $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$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}$
461Data not computed