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Properties

Label 20.0.20609556882...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{35}\cdot 11^{12}\cdot 41^{12}$
Root discriminant $654.19$
Ramified primes $5, 11, 41$
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![3005011477995800801, 0, 0, 0, 0, 146151013550157, 0, 0, 0, 0, -76999681, 0, 0, 0, 0, -25947, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 25947*x^15 - 76999681*x^10 + 146151013550157*x^5 + 3005011477995800801)
 
gp: K = bnfinit(x^20 - 25947*x^15 - 76999681*x^10 + 146151013550157*x^5 + 3005011477995800801, 1)
 

Normalized defining polynomial

\( x^{20} - 25947 x^{15} - 76999681 x^{10} + 146151013550157 x^{5} + 3005011477995800801 \)

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:  \(206095568825660384301069002308652852661907672882080078125=5^{35}\cdot 11^{12}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $654.19$
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:  $5, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{11} a^{7} + \frac{2}{11} a^{2}$, $\frac{1}{11} a^{8} + \frac{2}{11} a^{3}$, $\frac{1}{121} a^{9} - \frac{53}{121} a^{4}$, $\frac{1}{54571} a^{10} - \frac{25947}{54571} a^{5}$, $\frac{1}{54571} a^{11} - \frac{25947}{54571} a^{6}$, $\frac{1}{600281} a^{12} - \frac{25947}{600281} a^{7} - \frac{3}{11} a^{2}$, $\frac{1}{24611521} a^{13} - \frac{25947}{24611521} a^{8} - \frac{58}{451} a^{3}$, $\frac{1}{270726731} a^{14} - \frac{25947}{270726731} a^{9} - \frac{1411}{4961} a^{4}$, $\frac{1}{32509904346334966881319} a^{15} - \frac{13149266904267436}{1711047597175524572701} a^{10} + \frac{224761342911890200}{595735910031609589} a^{5} - \frac{70124864530}{266261276999}$, $\frac{1}{32509904346334966881319} a^{16} - \frac{13149266904267436}{1711047597175524572701} a^{11} + \frac{224761342911890200}{595735910031609589} a^{6} - \frac{70124864530}{266261276999} a$, $\frac{1}{357608947809684635694509} a^{17} - \frac{13149266904267436}{18821523568930770299711} a^{12} + \frac{224761342911890200}{6553095010347705479} a^{7} - \frac{602647418528}{2928874046989} a^{2}$, $\frac{1}{161281635462167770698223559} a^{18} - \frac{107212831646100529}{8488507129587777405169661} a^{13} - \frac{71605251874366434139}{2955445849666815171029} a^{8} - \frac{300412845319402}{1320922195192039} a^{3}$, $\frac{1}{1774097990083845477680459149} a^{19} - \frac{107212831646100529}{93373578425465551456866271} a^{14} - \frac{71605251874366434139}{32509904346334966881319} a^{9} - \frac{6905023821279597}{14530144147112429} a^{4}$

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:  \( -\frac{2078185935}{32509904346334966881319} a^{15} + \frac{5429489721752}{1711047597175524572701} a^{10} - \frac{48589240370630}{595735910031609589} a^{5} - \frac{1733153313486}{266261276999} \) (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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R $20$ R $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ $20$ R $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$41$41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.5$x^{5} - 53136$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.2$x^{5} + 246$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.0.1$x^{5} - x + 7$$1$$5$$0$$C_5$$[\ ]^{5}$