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Properties

Label 20.0.36487627390...3376.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{3}$
Root discriminant $21.29$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, 125, 90, -284, -155, 519, 754, -323, -431, 242, 375, 272, 121, 12, 72, 5, 18, -8, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 - 8*x^17 + 18*x^16 + 5*x^15 + 72*x^14 + 12*x^13 + 121*x^12 + 272*x^11 + 375*x^10 + 242*x^9 - 431*x^8 - 323*x^7 + 754*x^6 + 519*x^5 - 155*x^4 - 284*x^3 + 90*x^2 + 125*x + 43)
 
gp: K = bnfinit(x^20 + 3*x^18 - 8*x^17 + 18*x^16 + 5*x^15 + 72*x^14 + 12*x^13 + 121*x^12 + 272*x^11 + 375*x^10 + 242*x^9 - 431*x^8 - 323*x^7 + 754*x^6 + 519*x^5 - 155*x^4 - 284*x^3 + 90*x^2 + 125*x + 43, 1)
 

Normalized defining polynomial

\( x^{20} + 3 x^{18} - 8 x^{17} + 18 x^{16} + 5 x^{15} + 72 x^{14} + 12 x^{13} + 121 x^{12} + 272 x^{11} + 375 x^{10} + 242 x^{9} - 431 x^{8} - 323 x^{7} + 754 x^{6} + 519 x^{5} - 155 x^{4} - 284 x^{3} + 90 x^{2} + 125 x + 43 \)

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:  \(364876273903737477578853376=2^{10}\cdot 11^{17}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.29$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{196593611292632726422909462033} a^{19} - \frac{15144383212641523807025292658}{196593611292632726422909462033} a^{18} + \frac{96724885421321401077563755311}{196593611292632726422909462033} a^{17} - \frac{27500599746083966483784492384}{196593611292632726422909462033} a^{16} - \frac{38945239586072686154528097134}{196593611292632726422909462033} a^{15} - \frac{69164783869103730818722405571}{196593611292632726422909462033} a^{14} + \frac{67158401939131259024589515700}{196593611292632726422909462033} a^{13} - \frac{29678572919645115170375733036}{196593611292632726422909462033} a^{12} + \frac{4769615063441416051448038623}{196593611292632726422909462033} a^{11} + \frac{68916214385251927880796145469}{196593611292632726422909462033} a^{10} - \frac{22675298852694303542206555373}{196593611292632726422909462033} a^{9} + \frac{67920007103565278667698253546}{196593611292632726422909462033} a^{8} + \frac{2238174353878475412530963143}{196593611292632726422909462033} a^{7} + \frac{8764100200109150051360711008}{196593611292632726422909462033} a^{6} - \frac{33442038858887031129779041175}{196593611292632726422909462033} a^{5} - \frac{284857943724670497862562058}{196593611292632726422909462033} a^{4} - \frac{20623662432091741978305649464}{196593611292632726422909462033} a^{3} - \frac{21105372886801626064415925464}{196593611292632726422909462033} a^{2} + \frac{62657536638959968792280556047}{196593611292632726422909462033} a + \frac{59866301351737620894877506658}{196593611292632726422909462033}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 88772.5750871 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
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)$$[2, 2, 2, 2, 2]^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
89Data not computed