Properties

Label 20.20.144...000.1
Degree $20$
Signature $[20, 0]$
Discriminant $1.441\times 10^{38}$
Root discriminant $80.90$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^4:C_5$ (as 20T23)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 20*x^18 + 390*x^17 - 300*x^16 - 5210*x^15 + 9890*x^14 + 26870*x^13 - 82520*x^12 - 22800*x^11 + 257912*x^10 - 177160*x^9 - 222825*x^8 + 304530*x^7 - 17950*x^6 - 112348*x^5 + 38430*x^4 + 6000*x^3 - 3100*x^2 + 250*x - 5)
 
gp: K = bnfinit(x^20 - 10*x^19 - 20*x^18 + 390*x^17 - 300*x^16 - 5210*x^15 + 9890*x^14 + 26870*x^13 - 82520*x^12 - 22800*x^11 + 257912*x^10 - 177160*x^9 - 222825*x^8 + 304530*x^7 - 17950*x^6 - 112348*x^5 + 38430*x^4 + 6000*x^3 - 3100*x^2 + 250*x - 5, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 250, -3100, 6000, 38430, -112348, -17950, 304530, -222825, -177160, 257912, -22800, -82520, 26870, 9890, -5210, -300, 390, -20, -10, 1]);
 

\( x^{20} - 10 x^{19} - 20 x^{18} + 390 x^{17} - 300 x^{16} - 5210 x^{15} + 9890 x^{14} + 26870 x^{13} - 82520 x^{12} - 22800 x^{11} + 257912 x^{10} - 177160 x^{9} - 222825 x^{8} + 304530 x^{7} - 17950 x^{6} - 112348 x^{5} + 38430 x^{4} + 6000 x^{3} - 3100 x^{2} + 250 x - 5 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[20, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(144120025000000000000000000000000000000\)\(\medspace = 2^{30}\cdot 5^{32}\cdot 7^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $80.90$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{10} - \frac{2}{5} a^{5}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{11} - \frac{2}{5} a^{6}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{12} - \frac{2}{5} a^{7}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{13} - \frac{2}{5} a^{8}$, $\frac{1}{3372389841484724867871691957878755} a^{19} + \frac{46154845468847310512005499177979}{674477968296944973574338391575751} a^{18} + \frac{57784908435380719977628424556464}{3372389841484724867871691957878755} a^{17} - \frac{229059293481243205024299143241964}{3372389841484724867871691957878755} a^{16} + \frac{295156289906449800111704912602143}{3372389841484724867871691957878755} a^{15} + \frac{277922538326282028437280280875796}{3372389841484724867871691957878755} a^{14} - \frac{179888564440527496865024764615943}{674477968296944973574338391575751} a^{13} + \frac{816345249265982920686671098105074}{3372389841484724867871691957878755} a^{12} - \frac{117981756623392051542819910796209}{3372389841484724867871691957878755} a^{11} - \frac{417795774068408782329174125399837}{3372389841484724867871691957878755} a^{10} + \frac{299583803375031689259079992804263}{3372389841484724867871691957878755} a^{9} - \frac{25244125992345059326007452736555}{674477968296944973574338391575751} a^{8} - \frac{1347462809014102955342803311209313}{3372389841484724867871691957878755} a^{7} - \frac{612817481989439779448060776732112}{3372389841484724867871691957878755} a^{6} + \frac{1063674912451157910525672826148264}{3372389841484724867871691957878755} a^{5} - \frac{319301125641488197333088874804911}{674477968296944973574338391575751} a^{4} + \frac{276913404820806739446426172072980}{674477968296944973574338391575751} a^{3} - \frac{133293239998776282816461856737103}{674477968296944973574338391575751} a^{2} + \frac{73929518930840694829355913458398}{674477968296944973574338391575751} a + \frac{88731391841842828990646035076259}{674477968296944973574338391575751}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 15494663083400 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{20}\cdot(2\pi)^{0}\cdot 15494663083400 \cdot 1}{2\sqrt{144120025000000000000000000000000000000}}\approx 0.676690205636786$ (assuming GRH)

Galois group

$C_2^4:C_5$ (as 20T23):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 80
The 8 conjugacy class representatives for $C_2^4:C_5$
Character table for $C_2^4:C_5$

Intermediate fields

5.5.390625.1, 10.10.7656250000000000.2, 10.10.7656250000000000.1, 10.10.375156250000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$