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Properties

Label 20.20.4192558812...3264.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{40}\cdot 3^{10}\cdot 71^{8}$
Root discriminant $38.12$
Ramified primes $2, 3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{10}\times D_5$ (as 20T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, 0, -1840, 0, 7690, 0, -16156, 0, 19459, 0, -14324, 0, 6591, 0, -1876, 0, 315, 0, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 28*x^18 + 315*x^16 - 1876*x^14 + 6591*x^12 - 14324*x^10 + 19459*x^8 - 16156*x^6 + 7690*x^4 - 1840*x^2 + 169)
 
gp: K = bnfinit(x^20 - 28*x^18 + 315*x^16 - 1876*x^14 + 6591*x^12 - 14324*x^10 + 19459*x^8 - 16156*x^6 + 7690*x^4 - 1840*x^2 + 169, 1)
 

Normalized defining polynomial

\( x^{20} - 28 x^{18} + 315 x^{16} - 1876 x^{14} + 6591 x^{12} - 14324 x^{10} + 19459 x^{8} - 16156 x^{6} + 7690 x^{4} - 1840 x^{2} + 169 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41925588122943302709337877643264=2^{40}\cdot 3^{10}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.12$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} + \frac{5}{13} a^{15} - \frac{1}{13} a^{13} + \frac{2}{13} a^{11} + \frac{1}{13} a^{9} - \frac{4}{13} a^{7} - \frac{4}{13} a^{5} + \frac{1}{13} a^{3} + \frac{1}{13} a$, $\frac{1}{97747} a^{18} - \frac{3778}{97747} a^{16} - \frac{5500}{97747} a^{14} - \frac{1493}{97747} a^{12} + \frac{33762}{97747} a^{10} - \frac{39459}{97747} a^{8} + \frac{17}{949} a^{6} - \frac{33357}{97747} a^{4} - \frac{19720}{97747} a^{2} - \frac{3563}{7519}$, $\frac{1}{97747} a^{19} + \frac{3741}{97747} a^{17} + \frac{32095}{97747} a^{15} - \frac{9012}{97747} a^{13} + \frac{48800}{97747} a^{11} - \frac{31940}{97747} a^{9} - \frac{275}{949} a^{7} + \frac{34314}{97747} a^{5} - \frac{12201}{97747} a^{3} - \frac{38800}{97747} a$

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:  $19$
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:  \( 1771371001.17 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}\times D_5$ (as 20T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 40 conjugacy class representatives for $C_{10}\times D_5$
Character table for $C_{10}\times D_5$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 10.10.6323239406592.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
$71$71.5.4.4$x^{5} + 568$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.4.4$x^{5} + 568$$5$$1$$4$$C_5$$[\ ]_{5}$