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Properties

Label 20.0.53687091200...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{44}\cdot 5^{15}$
Root discriminant $15.36$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 10, 7, -27, -84, -98, 52, 302, 462, 282, -78, 17, -8, 32, 21, 8, 2, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 2*x^17 + 8*x^16 + 21*x^15 + 32*x^14 - 8*x^13 + 17*x^12 - 78*x^11 + 282*x^10 + 462*x^9 + 302*x^8 + 52*x^7 - 98*x^6 - 84*x^5 - 27*x^4 + 7*x^3 + 10*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 2*x^17 + 8*x^16 + 21*x^15 + 32*x^14 - 8*x^13 + 17*x^12 - 78*x^11 + 282*x^10 + 462*x^9 + 302*x^8 + 52*x^7 - 98*x^6 - 84*x^5 - 27*x^4 + 7*x^3 + 10*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 2 x^{17} + 8 x^{16} + 21 x^{15} + 32 x^{14} - 8 x^{13} + 17 x^{12} - 78 x^{11} + 282 x^{10} + 462 x^{9} + 302 x^{8} + 52 x^{7} - 98 x^{6} - 84 x^{5} - 27 x^{4} + 7 x^{3} + 10 x^{2} + 4 x + 1 \)

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:  \(536870912000000000000000=2^{44}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.36$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{9} + \frac{1}{5} a^{6} + \frac{1}{5} a^{3} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{10} + \frac{1}{5} a^{7} + \frac{1}{5} a^{4} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} + \frac{1}{5} a^{8} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} - \frac{1}{5}$, $\frac{1}{10} a^{16} - \frac{1}{10} a^{12} + \frac{2}{5} a^{9} + \frac{2}{5} a^{6} + \frac{2}{5} a^{3} + \frac{2}{5} a - \frac{1}{10}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{13} + \frac{2}{5} a^{10} + \frac{2}{5} a^{7} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{10} a$, $\frac{1}{21610} a^{18} + \frac{431}{10805} a^{17} + \frac{213}{4322} a^{16} + \frac{981}{10805} a^{15} - \frac{1863}{21610} a^{14} - \frac{1066}{10805} a^{13} + \frac{1027}{21610} a^{12} - \frac{629}{10805} a^{11} - \frac{346}{10805} a^{10} - \frac{2689}{10805} a^{9} - \frac{1584}{10805} a^{8} - \frac{2461}{10805} a^{7} - \frac{1624}{10805} a^{6} - \frac{4144}{10805} a^{5} - \frac{2171}{10805} a^{4} + \frac{3703}{10805} a^{3} - \frac{1485}{4322} a^{2} + \frac{4039}{10805} a - \frac{1611}{4322}$, $\frac{1}{50274412722110} a^{19} + \frac{441212326}{25137206361055} a^{18} - \frac{1190617006952}{25137206361055} a^{17} - \frac{1584128515441}{50274412722110} a^{16} + \frac{255919186951}{10054882544422} a^{15} + \frac{359401331043}{5027441272211} a^{14} + \frac{435549748343}{25137206361055} a^{13} - \frac{3314300944831}{50274412722110} a^{12} - \frac{2018073398323}{5027441272211} a^{11} - \frac{4198600735447}{25137206361055} a^{10} - \frac{7715244288243}{25137206361055} a^{9} - \frac{653227244142}{5027441272211} a^{8} - \frac{5080319750457}{25137206361055} a^{7} + \frac{9835937399922}{25137206361055} a^{6} - \frac{1665931383467}{5027441272211} a^{5} - \frac{843534631392}{5027441272211} a^{4} - \frac{11341196986073}{50274412722110} a^{3} + \frac{3565740720977}{25137206361055} a^{2} - \frac{11800394419379}{25137206361055} a - \frac{13215028701711}{50274412722110}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{12939321934576}{25137206361055} a^{19} - \frac{7519882436982}{25137206361055} a^{18} - \frac{1525222143982}{5027441272211} a^{17} + \frac{5033594500842}{5027441272211} a^{16} + \frac{119988710077212}{25137206361055} a^{15} + \frac{306886992689078}{25137206361055} a^{14} + \frac{505469903193998}{25137206361055} a^{13} + \frac{5688015974078}{25137206361055} a^{12} + \frac{64330806931688}{25137206361055} a^{11} - \frac{944295886411252}{25137206361055} a^{10} + \frac{3277034230701268}{25137206361055} a^{9} + \frac{7574105508294573}{25137206361055} a^{8} + \frac{6073450638362028}{25137206361055} a^{7} + \frac{238469425502808}{25137206361055} a^{6} - \frac{2150602951119312}{25137206361055} a^{5} - \frac{1661807888977958}{25137206361055} a^{4} - \frac{81214305417920}{5027441272211} a^{3} + \frac{257869297557998}{25137206361055} a^{2} + \frac{183180531707738}{25137206361055} a + \frac{61067582174651}{25137206361055} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23009.4273672 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.256000.1, 10.2.327680000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed