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Properties

Label 20.0.51520750781...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 641^{4}$
Root discriminant $12.18$
Ramified primes $5, 641$
Class number $1$
Class group Trivial
Galois group $D_5^2:C_4$ (as 20T93)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 17, -39, 69, -100, 124, -139, 139, -125, 107, -77, 60, -33, 28, -10, 11, -2, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 - 2*x^17 + 11*x^16 - 10*x^15 + 28*x^14 - 33*x^13 + 60*x^12 - 77*x^11 + 107*x^10 - 125*x^9 + 139*x^8 - 139*x^7 + 124*x^6 - 100*x^5 + 69*x^4 - 39*x^3 + 17*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 + 4*x^18 - 2*x^17 + 11*x^16 - 10*x^15 + 28*x^14 - 33*x^13 + 60*x^12 - 77*x^11 + 107*x^10 - 125*x^9 + 139*x^8 - 139*x^7 + 124*x^6 - 100*x^5 + 69*x^4 - 39*x^3 + 17*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} - 2 x^{17} + 11 x^{16} - 10 x^{15} + 28 x^{14} - 33 x^{13} + 60 x^{12} - 77 x^{11} + 107 x^{10} - 125 x^{9} + 139 x^{8} - 139 x^{7} + 124 x^{6} - 100 x^{5} + 69 x^{4} - 39 x^{3} + 17 x^{2} - 5 x + 1 \)

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:  \(5152075078155517578125=5^{15}\cdot 641^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.18$
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:  $5, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{43737} a^{19} - \frac{928}{14579} a^{18} - \frac{88}{717} a^{17} + \frac{345}{14579} a^{16} + \frac{5213}{43737} a^{15} - \frac{2299}{14579} a^{14} + \frac{733}{43737} a^{13} + \frac{355}{43737} a^{12} + \frac{5897}{14579} a^{11} + \frac{3540}{14579} a^{10} + \frac{62}{183} a^{9} + \frac{1711}{14579} a^{8} - \frac{5764}{14579} a^{7} + \frac{5170}{14579} a^{6} + \frac{17861}{43737} a^{5} - \frac{3578}{14579} a^{4} - \frac{3425}{43737} a^{3} + \frac{15074}{43737} a^{2} + \frac{21521}{43737} a + \frac{5221}{43737}$

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:  \( \frac{109415}{43737} a^{19} + \frac{31424}{43737} a^{18} + \frac{7004}{717} a^{17} - \frac{107200}{43737} a^{16} + \frac{1099603}{43737} a^{15} - \frac{769744}{43737} a^{14} + \frac{2670073}{43737} a^{13} - \frac{2736983}{43737} a^{12} + \frac{1790669}{14579} a^{11} - \frac{2163464}{14579} a^{10} + \frac{12641}{61} a^{9} - \frac{10117448}{43737} a^{8} + \frac{3678809}{14579} a^{7} - \frac{3532347}{14579} a^{6} + \frac{3009694}{14579} a^{5} - \frac{6886816}{43737} a^{4} + \frac{4409678}{43737} a^{3} - \frac{709698}{14579} a^{2} + \frac{751138}{43737} a - \frac{124453}{43737} \) (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:  \( 1274.32576906 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2:C_4$ (as 20T93):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $D_5^2:C_4$
Character table for $D_5^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.1284003125.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
641Data not computed