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Properties

Label 20.0.12846961306...0752.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{15}$
Root discriminant $143.03$
Ramified primes $2, 11, 23$
Class number $96$ (GRH)
Class group $[2, 2, 2, 12]$ (GRH)
Galois group 20T130

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![606618891407, 0, 69306541424, 0, -3637093477, 0, 79474844, 0, 160397561, 0, 3126919, 0, -286189, 0, 17986, 0, 782, 0, -69, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 69*x^18 + 782*x^16 + 17986*x^14 - 286189*x^12 + 3126919*x^10 + 160397561*x^8 + 79474844*x^6 - 3637093477*x^4 + 69306541424*x^2 + 606618891407)
 
gp: K = bnfinit(x^20 - 69*x^18 + 782*x^16 + 17986*x^14 - 286189*x^12 + 3126919*x^10 + 160397561*x^8 + 79474844*x^6 - 3637093477*x^4 + 69306541424*x^2 + 606618891407, 1)
 

Normalized defining polynomial

\( x^{20} - 69 x^{18} + 782 x^{16} + 17986 x^{14} - 286189 x^{12} + 3126919 x^{10} + 160397561 x^{8} + 79474844 x^{6} - 3637093477 x^{4} + 69306541424 x^{2} + 606618891407 \)

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:  \(12846961306316291046734327469906326168010752=2^{20}\cdot 11^{16}\cdot 23^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.03$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{23} a^{4}$, $\frac{1}{23} a^{5}$, $\frac{1}{23} a^{6}$, $\frac{1}{23} a^{7}$, $\frac{1}{529} a^{8}$, $\frac{1}{529} a^{9}$, $\frac{1}{529} a^{10}$, $\frac{1}{529} a^{11}$, $\frac{1}{12167} a^{12}$, $\frac{1}{12167} a^{13}$, $\frac{1}{12167} a^{14}$, $\frac{1}{12167} a^{15}$, $\frac{1}{279841} a^{16}$, $\frac{1}{279841} a^{17}$, $\frac{1}{50665415279246700329259190803617} a^{18} - \frac{3127884780840376138544134}{2202844142575943492576486556679} a^{16} - \frac{20965660536367832013983481}{2202844142575943492576486556679} a^{14} + \frac{3741139398977616436483950}{95775832285910586633760285073} a^{12} + \frac{31760922878227436907110386}{95775832285910586633760285073} a^{10} - \frac{162946612505617325943092}{181050722657675967171569537} a^{8} - \frac{50907762691243949400244746}{4164166621126547244946099351} a^{6} - \frac{2009750094369503199133334}{181050722657675967171569537} a^{4} + \frac{10491823948566502144368922}{181050722657675967171569537} a^{2} + \frac{616933739960506638375989}{7871770550333737703111719}$, $\frac{1}{15554282490728737001082571576710419} a^{19} - \frac{33334331659270127536538299}{29403180511774550096564407517411} a^{17} + \frac{20437765999781016458373374200}{676273151770814652220981372900453} a^{15} + \frac{523277995721004304841857404}{29403180511774550096564407517411} a^{13} + \frac{5101181157293154517711057422}{29403180511774550096564407517411} a^{11} - \frac{1003462631980013886791879429}{1278399152685850004198452500757} a^{9} + \frac{15519454385868889227354735436}{1278399152685850004198452500757} a^{7} + \frac{580501270630327086831133872}{55582571855906521921671847859} a^{5} + \frac{17934513367058487252129753085}{55582571855906521921671847859} a^{3} + \frac{39975786491629195153934584}{2416633558952457474855297733} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{12}$, which has order $96$ (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:  \( 132268587065 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T130:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n130
Character table for t20n130 is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\zeta_{11})^+\), 10.0.1379687283212183.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.10.10.13$x^{10} - 15 x^{8} + 26 x^{6} - 22 x^{4} + 37 x^{2} - 59$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$