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Properties

Label 20.0.78125000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 5^{31}$
Root discriminant $27.84$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_5:F_5$ (as 20T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -80, 400, -710, 225, 516, -1210, 200, 2080, -200, -1754, 60, 1025, 20, -400, -4, 105, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 105*x^16 - 4*x^15 - 400*x^14 + 20*x^13 + 1025*x^12 + 60*x^11 - 1754*x^10 - 200*x^9 + 2080*x^8 + 200*x^7 - 1210*x^6 + 516*x^5 + 225*x^4 - 710*x^3 + 400*x^2 - 80*x + 16)
 
gp: K = bnfinit(x^20 - 15*x^18 + 105*x^16 - 4*x^15 - 400*x^14 + 20*x^13 + 1025*x^12 + 60*x^11 - 1754*x^10 - 200*x^9 + 2080*x^8 + 200*x^7 - 1210*x^6 + 516*x^5 + 225*x^4 - 710*x^3 + 400*x^2 - 80*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} + 105 x^{16} - 4 x^{15} - 400 x^{14} + 20 x^{13} + 1025 x^{12} + 60 x^{11} - 1754 x^{10} - 200 x^{9} + 2080 x^{8} + 200 x^{7} - 1210 x^{6} + 516 x^{5} + 225 x^{4} - 710 x^{3} + 400 x^{2} - 80 x + 16 \)

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:  \(78125000000000000000000000000=2^{24}\cdot 5^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.84$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{12} - \frac{2}{25} a^{11} - \frac{1}{25} a^{10} + \frac{1}{5} a^{9} - \frac{2}{25} a^{8} - \frac{9}{25} a^{7} - \frac{6}{25} a^{6} + \frac{12}{25} a^{5} - \frac{1}{5} a^{4} - \frac{4}{25} a^{3} - \frac{3}{25} a^{2} - \frac{7}{25} a - \frac{6}{25}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{12} - \frac{2}{25} a^{11} + \frac{2}{25} a^{10} - \frac{12}{25} a^{9} - \frac{1}{5} a^{8} + \frac{2}{25} a^{7} + \frac{9}{25} a^{6} + \frac{6}{25} a^{5} + \frac{6}{25} a^{4} + \frac{1}{5} a^{3} + \frac{4}{25} a^{2} + \frac{3}{25} a + \frac{7}{25}$, $\frac{1}{25} a^{15} + \frac{2}{25} a^{10} - \frac{12}{25} a^{5} + \frac{9}{25}$, $\frac{1}{25} a^{16} + \frac{2}{25} a^{11} - \frac{12}{25} a^{6} + \frac{9}{25} a$, $\frac{1}{50} a^{17} - \frac{1}{50} a^{15} - \frac{1}{50} a^{13} + \frac{1}{25} a^{11} + \frac{2}{25} a^{10} - \frac{1}{10} a^{9} + \frac{1}{25} a^{8} + \frac{11}{25} a^{7} + \frac{3}{25} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{25} a^{3} + \frac{6}{25} a^{2} + \frac{7}{50} a + \frac{1}{25}$, $\frac{1}{1100} a^{18} - \frac{2}{275} a^{17} - \frac{3}{220} a^{16} + \frac{4}{275} a^{15} + \frac{13}{1100} a^{14} + \frac{3}{275} a^{13} + \frac{9}{275} a^{12} + \frac{16}{275} a^{11} + \frac{13}{1100} a^{10} - \frac{89}{275} a^{9} - \frac{111}{550} a^{8} + \frac{73}{275} a^{7} + \frac{133}{275} a^{6} - \frac{131}{275} a^{5} - \frac{111}{550} a^{4} - \frac{129}{275} a^{3} + \frac{69}{220} a^{2} + \frac{37}{550} a - \frac{18}{55}$, $\frac{1}{54640555298936200} a^{19} + \frac{501213044843}{2732027764946810} a^{18} + \frac{55659507015377}{54640555298936200} a^{17} - \frac{170141821283857}{13660138824734050} a^{16} + \frac{112044023453849}{54640555298936200} a^{15} - \frac{44996971234654}{6830069412367025} a^{14} - \frac{126585345145219}{6830069412367025} a^{13} - \frac{424417714823489}{13660138824734050} a^{12} + \frac{1535700004042441}{54640555298936200} a^{11} - \frac{1214652867671}{6830069412367025} a^{10} + \frac{817912004738843}{27320277649468100} a^{9} + \frac{913427292277218}{6830069412367025} a^{8} - \frac{2522812844562418}{6830069412367025} a^{7} - \frac{198165917231651}{6830069412367025} a^{6} + \frac{1128672173413967}{27320277649468100} a^{5} + \frac{1165145643477797}{2732027764946810} a^{4} - \frac{5053713645226167}{54640555298936200} a^{3} + \frac{1054236916723409}{2483661604497100} a^{2} + \frac{307123960095464}{1366013882473405} a + \frac{457067848171876}{1366013882473405}$

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:  \( -\frac{2983543014786833}{54640555298936200} a^{19} - \frac{148784274151926}{6830069412367025} a^{18} + \frac{44079302694659107}{54640555298936200} a^{17} + \frac{2181800482638006}{6830069412367025} a^{16} - \frac{303527392473747157}{54640555298936200} a^{15} - \frac{26779039769557259}{13660138824734050} a^{14} + \frac{283175272197727821}{13660138824734050} a^{13} + \frac{94589353765057731}{13660138824734050} a^{12} - \frac{2846489988603959249}{54640555298936200} a^{11} - \frac{315166558838092591}{13660138824734050} a^{10} + \frac{2296093435495594943}{27320277649468100} a^{9} + \frac{283631413387447087}{6830069412367025} a^{8} - \frac{640660990877437163}{6830069412367025} a^{7} - \frac{296382860912163892}{6830069412367025} a^{6} + \frac{1270966774142201717}{27320277649468100} a^{5} - \frac{202428080077376873}{13660138824734050} a^{4} - \frac{1053134780796481897}{54640555298936200} a^{3} + \frac{78340854195522009}{2483661604497100} a^{2} - \frac{132187998922937747}{13660138824734050} a + \frac{11102013853539689}{6830069412367025} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14787405.0482 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_5:F_5$ (as 20T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 20 conjugacy class representatives for $C_2\times C_5:F_5$
Character table for $C_2\times C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.125000000000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed