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Properties

Label 20.12.1806627930...5625.2
Degree $20$
Signature $[12, 4]$
Discriminant $3^{16}\cdot 5^{10}\cdot 73^{10}$
Root discriminant $46.01$
Ramified primes $3, 5, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![829, 7623, 23377, 41674, 37862, -77692, -150225, 82145, 112870, -65409, -13815, 18915, -10259, 2205, 2095, -1238, 8, 128, -21, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 21*x^18 + 128*x^17 + 8*x^16 - 1238*x^15 + 2095*x^14 + 2205*x^13 - 10259*x^12 + 18915*x^11 - 13815*x^10 - 65409*x^9 + 112870*x^8 + 82145*x^7 - 150225*x^6 - 77692*x^5 + 37862*x^4 + 41674*x^3 + 23377*x^2 + 7623*x + 829)
 
gp: K = bnfinit(x^20 - 4*x^19 - 21*x^18 + 128*x^17 + 8*x^16 - 1238*x^15 + 2095*x^14 + 2205*x^13 - 10259*x^12 + 18915*x^11 - 13815*x^10 - 65409*x^9 + 112870*x^8 + 82145*x^7 - 150225*x^6 - 77692*x^5 + 37862*x^4 + 41674*x^3 + 23377*x^2 + 7623*x + 829, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 21 x^{18} + 128 x^{17} + 8 x^{16} - 1238 x^{15} + 2095 x^{14} + 2205 x^{13} - 10259 x^{12} + 18915 x^{11} - 13815 x^{10} - 65409 x^{9} + 112870 x^{8} + 82145 x^{7} - 150225 x^{6} - 77692 x^{5} + 37862 x^{4} + 41674 x^{3} + 23377 x^{2} + 7623 x + 829 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1806627930211353113514833291015625=3^{16}\cdot 5^{10}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.01$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{11} + \frac{1}{15} a^{10} - \frac{1}{5} a^{9} - \frac{1}{3} a^{8} + \frac{2}{15} a^{7} + \frac{2}{15} a^{6} - \frac{1}{5} a^{5} + \frac{2}{15} a^{4} + \frac{4}{15} a^{3} + \frac{1}{15} a^{2} - \frac{2}{5} a - \frac{7}{15}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{10} - \frac{2}{15} a^{9} + \frac{1}{15} a^{8} - \frac{1}{5} a^{7} + \frac{1}{15} a^{6} - \frac{1}{15} a^{5} - \frac{4}{15} a^{4} - \frac{1}{5} a^{3} + \frac{2}{15} a^{2} + \frac{2}{15} a - \frac{1}{3}$, $\frac{1}{15} a^{14} - \frac{1}{15} a^{11} + \frac{1}{15} a^{10} + \frac{1}{15} a^{9} + \frac{2}{5} a^{8} - \frac{2}{15} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{2}{5} a^{4} + \frac{2}{15} a^{3} - \frac{4}{15} a^{2} - \frac{2}{15} a + \frac{1}{5}$, $\frac{1}{15} a^{15} - \frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{2}{5} a^{9} + \frac{2}{15} a^{8} + \frac{4}{15} a^{7} + \frac{7}{15} a^{6} - \frac{1}{3} a^{4} + \frac{2}{5} a^{3} + \frac{2}{15} a^{2} + \frac{2}{5} a + \frac{1}{3}$, $\frac{1}{75} a^{16} + \frac{2}{75} a^{15} + \frac{2}{75} a^{13} + \frac{1}{75} a^{12} - \frac{1}{75} a^{11} - \frac{1}{15} a^{10} - \frac{4}{15} a^{9} - \frac{7}{25} a^{8} + \frac{2}{15} a^{7} - \frac{4}{75} a^{6} + \frac{11}{75} a^{5} - \frac{29}{75} a^{4} + \frac{1}{75} a^{3} - \frac{1}{15} a^{2} - \frac{11}{25} a + \frac{34}{75}$, $\frac{1}{75} a^{17} + \frac{1}{75} a^{15} + \frac{2}{75} a^{14} + \frac{2}{75} a^{13} + \frac{2}{75} a^{12} - \frac{1}{25} a^{11} - \frac{12}{25} a^{9} + \frac{4}{25} a^{8} - \frac{8}{25} a^{7} + \frac{8}{25} a^{6} - \frac{26}{75} a^{5} - \frac{2}{25} a^{4} + \frac{28}{75} a^{3} - \frac{28}{75} a^{2} - \frac{1}{3} a - \frac{13}{75}$, $\frac{1}{231375} a^{18} + \frac{194}{231375} a^{17} + \frac{11}{46275} a^{16} - \frac{2247}{77125} a^{15} - \frac{683}{46275} a^{14} - \frac{6182}{231375} a^{13} + \frac{2099}{231375} a^{12} - \frac{3547}{77125} a^{11} + \frac{5478}{77125} a^{10} - \frac{28659}{77125} a^{9} - \frac{2269}{15425} a^{8} + \frac{25161}{77125} a^{7} + \frac{14989}{231375} a^{6} + \frac{40379}{231375} a^{5} - \frac{43562}{231375} a^{4} + \frac{26796}{77125} a^{3} - \frac{20407}{231375} a^{2} + \frac{509}{46275} a + \frac{31289}{231375}$, $\frac{1}{315709952068327424387739552804375} a^{19} - \frac{32031785684301334245052189}{63141990413665484877547910560875} a^{18} + \frac{1060085040586593818742673212689}{315709952068327424387739552804375} a^{17} - \frac{54770331220754846668916700579}{35078883563147491598637728089375} a^{16} + \frac{10014103760169912911662664540159}{315709952068327424387739552804375} a^{15} - \frac{2683243272195348093045978574822}{315709952068327424387739552804375} a^{14} + \frac{6876740997905640548822688051347}{315709952068327424387739552804375} a^{13} + \frac{5277196316823183405452124273073}{315709952068327424387739552804375} a^{12} - \frac{6044082347631468755724837999764}{105236650689442474795913184268125} a^{11} - \frac{1593241675549050189150319509101}{105236650689442474795913184268125} a^{10} - \frac{15250236136706463788227583609044}{105236650689442474795913184268125} a^{9} + \frac{2355383817767059199307162566}{105236650689442474795913184268125} a^{8} - \frac{77528681687192800782706300072148}{315709952068327424387739552804375} a^{7} + \frac{34231616004418257052656903241183}{315709952068327424387739552804375} a^{6} + \frac{42684705660304234793321057429057}{315709952068327424387739552804375} a^{5} + \frac{16845586669898471749204096396459}{35078883563147491598637728089375} a^{4} - \frac{14214531365344542602420685221014}{315709952068327424387739552804375} a^{3} - \frac{39644946808899944169064189642432}{315709952068327424387739552804375} a^{2} + \frac{1436080405499588318631232968689}{10184192002204110464120630735625} a + \frac{63319671055744784498738038112404}{315709952068327424387739552804375}$

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:  $15$
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:  \( 4787053745.47 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.10791225.1, 10.10.582252685003125.1, 10.6.42504446005228125.2, 10.6.8500889201045625.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$