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Properties

Label 20.16.4115394215...0000.2
Degree $20$
Signature $[16, 2]$
Discriminant $2^{24}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $42.73$
Ramified primes $2, 5, 3469$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 205, -742, -2011, 4228, 6410, -10148, -9569, 13660, 7507, -11122, -2993, 5384, 372, -1410, 99, 158, -21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 21*x^18 + 158*x^17 + 99*x^16 - 1410*x^15 + 372*x^14 + 5384*x^13 - 2993*x^12 - 11122*x^11 + 7507*x^10 + 13660*x^9 - 9569*x^8 - 10148*x^7 + 6410*x^6 + 4228*x^5 - 2011*x^4 - 742*x^3 + 205*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^20 - 6*x^19 - 21*x^18 + 158*x^17 + 99*x^16 - 1410*x^15 + 372*x^14 + 5384*x^13 - 2993*x^12 - 11122*x^11 + 7507*x^10 + 13660*x^9 - 9569*x^8 - 10148*x^7 + 6410*x^6 + 4228*x^5 - 2011*x^4 - 742*x^3 + 205*x^2 - 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 21 x^{18} + 158 x^{17} + 99 x^{16} - 1410 x^{15} + 372 x^{14} + 5384 x^{13} - 2993 x^{12} - 11122 x^{11} + 7507 x^{10} + 13660 x^{9} - 9569 x^{8} - 10148 x^{7} + 6410 x^{6} + 4228 x^{5} - 2011 x^{4} - 742 x^{3} + 205 x^{2} - 2 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(411539421581712055500800000000000=2^{24}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.73$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2216106135256943855476063} a^{19} - \frac{461996993690973164571672}{2216106135256943855476063} a^{18} + \frac{988430313418273147450654}{2216106135256943855476063} a^{17} + \frac{976965104848872006570270}{2216106135256943855476063} a^{16} - \frac{72377917946752238724348}{2216106135256943855476063} a^{15} + \frac{609734386671201747055072}{2216106135256943855476063} a^{14} - \frac{885863035007628206644520}{2216106135256943855476063} a^{13} + \frac{672767110111798383798674}{2216106135256943855476063} a^{12} - \frac{388423047046045013823146}{2216106135256943855476063} a^{11} - \frac{788434904370144408746737}{2216106135256943855476063} a^{10} - \frac{309291115905395269518520}{2216106135256943855476063} a^{9} - \frac{1098695832556841223337642}{2216106135256943855476063} a^{8} + \frac{16837449188442966986656}{2216106135256943855476063} a^{7} + \frac{48717953389303187023849}{2216106135256943855476063} a^{6} + \frac{783236626660688068550662}{2216106135256943855476063} a^{5} + \frac{395122440480932985035281}{2216106135256943855476063} a^{4} + \frac{691957096297634181036729}{2216106135256943855476063} a^{3} + \frac{697701699107581576437779}{2216106135256943855476063} a^{2} - \frac{780503185236543048291391}{2216106135256943855476063} a + \frac{282265690100685446269803}{2216106135256943855476063}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $17$
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:  \( 2315872508.73 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed