↑

Properties

Label 20.0.42143994095...9025.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{2}\cdot 7^{2}\cdot 17^{12}$
Root discriminant $13.53$
Ramified primes $3, 5, 7, 17$
Class number $1$
Class group Trivial
Galois group 20T196

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 16, -33, 52, -61, 60, -50, 35, -35, 53, -45, 37, -27, 11, -5, 10, -8, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 5*x^18 - 8*x^17 + 10*x^16 - 5*x^15 + 11*x^14 - 27*x^13 + 37*x^12 - 45*x^11 + 53*x^10 - 35*x^9 + 35*x^8 - 50*x^7 + 60*x^6 - 61*x^5 + 52*x^4 - 33*x^3 + 16*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 5*x^18 - 8*x^17 + 10*x^16 - 5*x^15 + 11*x^14 - 27*x^13 + 37*x^12 - 45*x^11 + 53*x^10 - 35*x^9 + 35*x^8 - 50*x^7 + 60*x^6 - 61*x^5 + 52*x^4 - 33*x^3 + 16*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 5 x^{18} - 8 x^{17} + 10 x^{16} - 5 x^{15} + 11 x^{14} - 27 x^{13} + 37 x^{12} - 45 x^{11} + 53 x^{10} - 35 x^{9} + 35 x^{8} - 50 x^{7} + 60 x^{6} - 61 x^{5} + 52 x^{4} - 33 x^{3} + 16 x^{2} - 5 x + 1 \)

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:  \(42143994095570692679025=3^{10}\cdot 5^{2}\cdot 7^{2}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.53$
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:  $3, 5, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{20979369} a^{19} + \frac{2760670}{20979369} a^{18} - \frac{371684}{6993123} a^{17} + \frac{1870243}{20979369} a^{16} + \frac{1745804}{20979369} a^{15} - \frac{1155802}{6993123} a^{14} + \frac{1452494}{20979369} a^{13} - \frac{2752888}{20979369} a^{12} + \frac{199836}{2331041} a^{11} - \frac{554254}{6993123} a^{10} - \frac{276589}{20979369} a^{9} - \frac{2223436}{6993123} a^{8} + \frac{4038779}{20979369} a^{7} - \frac{3416210}{6993123} a^{6} - \frac{380514}{2331041} a^{5} + \frac{10175261}{20979369} a^{4} + \frac{1473991}{6993123} a^{3} + \frac{2123872}{6993123} a^{2} - \frac{879899}{20979369} a - \frac{3186121}{20979369}$

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:  \( \frac{3771380}{6993123} a^{19} - \frac{3350013}{2331041} a^{18} + \frac{5239348}{2331041} a^{17} - \frac{25323248}{6993123} a^{16} + \frac{30019405}{6993123} a^{15} - \frac{3463195}{2331041} a^{14} + \frac{39727114}{6993123} a^{13} - \frac{29703918}{2331041} a^{12} + \frac{37539115}{2331041} a^{11} - \frac{134329763}{6993123} a^{10} + \frac{158487140}{6993123} a^{9} - \frac{28879446}{2331041} a^{8} + \frac{38135956}{2331041} a^{7} - \frac{154049537}{6993123} a^{6} + \frac{59906801}{2331041} a^{5} - \frac{57705200}{2331041} a^{4} + \frac{144072433}{6993123} a^{3} - \frac{27335414}{2331041} a^{2} + \frac{43470103}{6993123} a - \frac{6552893}{6993123} \) (order $6$)
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:  \( 5347.89205383 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T196:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1280
The 44 conjugacy class representatives for t20n196
Character table for t20n196 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.44217.1, 10.0.5865429267.1, 10.0.68430008115.1, 10.2.205290024345.1

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 R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$