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Properties

Label 20.0.23760380172...2704.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 38569^{4}$
Root discriminant $23.38$
Ramified primes $2, 38569$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T279

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 50, -132, 197, -124, 102, -330, 652, -150, 54, -182, 411, -62, 2, -8, 51, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^17 + 51*x^16 - 8*x^15 + 2*x^14 - 62*x^13 + 411*x^12 - 182*x^11 + 54*x^10 - 150*x^9 + 652*x^8 - 330*x^7 + 102*x^6 - 124*x^5 + 197*x^4 - 132*x^3 + 50*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^17 + 51*x^16 - 8*x^15 + 2*x^14 - 62*x^13 + 411*x^12 - 182*x^11 + 54*x^10 - 150*x^9 + 652*x^8 - 330*x^7 + 102*x^6 - 124*x^5 + 197*x^4 - 132*x^3 + 50*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{17} + 51 x^{16} - 8 x^{15} + 2 x^{14} - 62 x^{13} + 411 x^{12} - 182 x^{11} + 54 x^{10} - 150 x^{9} + 652 x^{8} - 330 x^{7} + 102 x^{6} - 124 x^{5} + 197 x^{4} - 132 x^{3} + 50 x^{2} - 10 x + 1 \)

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:  \(2376038017203857413535432704=2^{30}\cdot 38569^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.38$
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:  $2, 38569$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{528775004165597208503} a^{19} - \frac{100039733464952041615}{528775004165597208503} a^{18} - \frac{261516655798890729877}{528775004165597208503} a^{17} - \frac{773983681308783910}{528775004165597208503} a^{16} + \frac{53373038632503929982}{528775004165597208503} a^{15} + \frac{216313679376519159431}{528775004165597208503} a^{14} + \frac{53086023854255865309}{528775004165597208503} a^{13} + \frac{171564166303696987388}{528775004165597208503} a^{12} + \frac{157874924476531773338}{528775004165597208503} a^{11} - \frac{165094087223348456148}{528775004165597208503} a^{10} + \frac{169017419050675412049}{528775004165597208503} a^{9} + \frac{35910474181721294909}{528775004165597208503} a^{8} - \frac{239298524198307216446}{528775004165597208503} a^{7} + \frac{2144576494504377733}{528775004165597208503} a^{6} - \frac{261479824632136117299}{528775004165597208503} a^{5} - \frac{141311803785371748715}{528775004165597208503} a^{4} + \frac{80181851617636570980}{528775004165597208503} a^{3} + \frac{28739848136638323746}{528775004165597208503} a^{2} - \frac{78139359976593654957}{528775004165597208503} a + \frac{170918509132450229025}{528775004165597208503}$

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

Class group and class number

$C_{4}$, 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:  \( \frac{253674170514925972148}{528775004165597208503} a^{19} + \frac{52542436516689621666}{528775004165597208503} a^{18} + \frac{18753273834318634080}{528775004165597208503} a^{17} - \frac{500364504880230745008}{528775004165597208503} a^{16} + \frac{12834302413375623670998}{528775004165597208503} a^{15} + \frac{614432521127988492913}{528775004165597208503} a^{14} + \frac{1030373414196219590801}{528775004165597208503} a^{13} - \frac{15420247827636877721447}{528775004165597208503} a^{12} + \frac{101083197269610258646448}{528775004165597208503} a^{11} - \frac{25654671351925011540596}{528775004165597208503} a^{10} + \frac{11447257858588476336666}{528775004165597208503} a^{9} - \frac{35868565992552923027873}{528775004165597208503} a^{8} + \frac{157985240307856513150405}{528775004165597208503} a^{7} - \frac{51595181434190083474168}{528775004165597208503} a^{6} + \frac{19886461314042749747235}{528775004165597208503} a^{5} - \frac{27984409354716143770313}{528775004165597208503} a^{4} + \frac{44171447199038627482930}{528775004165597208503} a^{3} - \frac{24099380587501576424047}{528775004165597208503} a^{2} + \frac{8808546018540881018497}{528775004165597208503} a - \frac{1197905272523560994916}{528775004165597208503} \) (order $4$)
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:  \( 110404.412177 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T279:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.38569.1, 10.0.1523269387264.2, 10.0.1523269387264.1, 10.10.1523269387264.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
38569Data not computed