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Properties

Label 20.0.25187809928...1601.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 37^{2}\cdot 47^{2}\cdot 239^{2}\cdot 1571417^{2}$
Root discriminant $26.31$
Ramified primes $3, 37, 47, 239, 1571417$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 17, 10, 91, 15, 362, -124, 704, -201, 830, -298, 574, -148, 235, -54, 65, -9, 10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 10*x^18 - 9*x^17 + 65*x^16 - 54*x^15 + 235*x^14 - 148*x^13 + 574*x^12 - 298*x^11 + 830*x^10 - 201*x^9 + 704*x^8 - 124*x^7 + 362*x^6 + 15*x^5 + 91*x^4 + 10*x^3 + 17*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 10*x^18 - 9*x^17 + 65*x^16 - 54*x^15 + 235*x^14 - 148*x^13 + 574*x^12 - 298*x^11 + 830*x^10 - 201*x^9 + 704*x^8 - 124*x^7 + 362*x^6 + 15*x^5 + 91*x^4 + 10*x^3 + 17*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 10 x^{18} - 9 x^{17} + 65 x^{16} - 54 x^{15} + 235 x^{14} - 148 x^{13} + 574 x^{12} - 298 x^{11} + 830 x^{10} - 201 x^{9} + 704 x^{8} - 124 x^{7} + 362 x^{6} + 15 x^{5} + 91 x^{4} + 10 x^{3} + 17 x^{2} + 3 x + 1 \)

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:  \(25187809928912253401368511601=3^{10}\cdot 37^{2}\cdot 47^{2}\cdot 239^{2}\cdot 1571417^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $3, 37, 47, 239, 1571417$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3267034799702099711} a^{19} - \frac{1510844052793069891}{3267034799702099711} a^{18} - \frac{250619487270919874}{3267034799702099711} a^{17} - \frac{257140762234279085}{3267034799702099711} a^{16} - \frac{152037829593508823}{3267034799702099711} a^{15} + \frac{414534774692684685}{3267034799702099711} a^{14} + \frac{862270185961993486}{3267034799702099711} a^{13} - \frac{1005423791078703117}{3267034799702099711} a^{12} + \frac{167220725391160959}{3267034799702099711} a^{11} + \frac{1020801247308587379}{3267034799702099711} a^{10} + \frac{666914700939533569}{3267034799702099711} a^{9} + \frac{1562005008014786457}{3267034799702099711} a^{8} - \frac{916055078781509513}{3267034799702099711} a^{7} + \frac{473718284670442967}{3267034799702099711} a^{6} - \frac{1344083310886696272}{3267034799702099711} a^{5} + \frac{358191077728910858}{3267034799702099711} a^{4} + \frac{91189459085091029}{3267034799702099711} a^{3} - \frac{1042161416139681591}{3267034799702099711} a^{2} + \frac{346337070942451268}{3267034799702099711} a + \frac{95406066622047799}{3267034799702099711}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( -\frac{945785585796613510}{3267034799702099711} a^{19} + \frac{981023347059747199}{3267034799702099711} a^{18} - \frac{9389066635156684202}{3267034799702099711} a^{17} + \frac{8790504739687032894}{3267034799702099711} a^{16} - \frac{60812168848393328889}{3267034799702099711} a^{15} + \frac{52750389587814447879}{3267034799702099711} a^{14} - \frac{217942589557688036339}{3267034799702099711} a^{13} + \frac{144784364466146292536}{3267034799702099711} a^{12} - \frac{526986067604205769691}{3267034799702099711} a^{11} + \frac{294831502881564351342}{3267034799702099711} a^{10} - \frac{745966809589009773739}{3267034799702099711} a^{9} + \frac{208013222996663422649}{3267034799702099711} a^{8} - \frac{608817540728773583124}{3267034799702099711} a^{7} + \frac{150488308944985198880}{3267034799702099711} a^{6} - \frac{296195506937508113675}{3267034799702099711} a^{5} + \frac{6393766537572833624}{3267034799702099711} a^{4} - \frac{64412078331137061641}{3267034799702099711} a^{3} + \frac{8358866792004002978}{3267034799702099711} a^{2} - \frac{10928284102620181358}{3267034799702099711} a + \frac{1516673687840527155}{3267034799702099711} \) (order $6$)
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:  \( 135507.566191 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1021:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.10.653113904957.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.7.0.1$x^{7} - 4 x + 5$$1$$7$$0$$C_7$$[\ ]^{7}$
37.7.0.1$x^{7} - 4 x + 5$$1$$7$$0$$C_7$$[\ ]^{7}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.8.0.1$x^{8} - x + 20$$1$$8$$0$$C_8$$[\ ]^{8}$
47.8.0.1$x^{8} - x + 20$$1$$8$$0$$C_8$$[\ ]^{8}$
239Data not computed
1571417Data not computed