↑

Properties

Label 20.12.6364883959...1904.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 13^{15}\cdot 17^{9}$
Root discriminant $49.00$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, 5928, -40560, 34060, 125736, -111956, -90012, 58708, 35984, 11700, -32760, 4108, 2246, 1392, -478, -148, -54, 40, 12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 12*x^18 + 40*x^17 - 54*x^16 - 148*x^15 - 478*x^14 + 1392*x^13 + 2246*x^12 + 4108*x^11 - 32760*x^10 + 11700*x^9 + 35984*x^8 + 58708*x^7 - 90012*x^6 - 111956*x^5 + 125736*x^4 + 34060*x^3 - 40560*x^2 + 5928*x - 52)
 
gp: K = bnfinit(x^20 - 8*x^19 + 12*x^18 + 40*x^17 - 54*x^16 - 148*x^15 - 478*x^14 + 1392*x^13 + 2246*x^12 + 4108*x^11 - 32760*x^10 + 11700*x^9 + 35984*x^8 + 58708*x^7 - 90012*x^6 - 111956*x^5 + 125736*x^4 + 34060*x^3 - 40560*x^2 + 5928*x - 52, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 12 x^{18} + 40 x^{17} - 54 x^{16} - 148 x^{15} - 478 x^{14} + 1392 x^{13} + 2246 x^{12} + 4108 x^{11} - 32760 x^{10} + 11700 x^{9} + 35984 x^{8} + 58708 x^{7} - 90012 x^{6} - 111956 x^{5} + 125736 x^{4} + 34060 x^{3} - 40560 x^{2} + 5928 x - 52 \)

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:  \(6364883959565411703183364041211904=2^{20}\cdot 13^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.00$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{52} a^{18} - \frac{3}{26} a^{17} + \frac{3}{13} a^{16} - \frac{2}{13} a^{15} - \frac{1}{13} a^{14} + \frac{2}{13} a^{13} + \frac{5}{26} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{20380096613170600996378192500677465587986716} a^{19} + \frac{187125137483587795091248555469873395656997}{20380096613170600996378192500677465587986716} a^{18} + \frac{702009496750910388608765212280754623955137}{5095024153292650249094548125169366396996679} a^{17} + \frac{127193464541740736597303863885373199391345}{536318331925542131483636644754670147052282} a^{16} - \frac{2254190831638377322431835690907199378416577}{10190048306585300498189096250338732793993358} a^{15} - \frac{1573131615353626197043194260418861061289211}{10190048306585300498189096250338732793993358} a^{14} - \frac{96375028290933454871877511880848473404964}{5095024153292650249094548125169366396996679} a^{13} + \frac{512732847580135311624432332972746203082868}{5095024153292650249094548125169366396996679} a^{12} + \frac{35048592299847043741957820700612952511897}{783849869737330807553007403872210214922566} a^{11} - \frac{77896164873028077320972778865642813520357}{783849869737330807553007403872210214922566} a^{10} + \frac{4395998152576650328366086532823826634401}{391924934868665403776503701936105107461283} a^{9} + \frac{262505015432974798990125867912627739946667}{783849869737330807553007403872210214922566} a^{8} + \frac{83105853568057035421859653379424098898475}{391924934868665403776503701936105107461283} a^{7} - \frac{113999952538287961372319573456627518140214}{391924934868665403776503701936105107461283} a^{6} - \frac{169704321149145012907112879711801391629813}{391924934868665403776503701936105107461283} a^{5} + \frac{102373354137176066865018666175766795375621}{391924934868665403776503701936105107461283} a^{4} + \frac{127605546313536280660775084244161365834903}{391924934868665403776503701936105107461283} a^{3} + \frac{136675423245025478578902865865300622785107}{391924934868665403776503701936105107461283} a^{2} + \frac{148127825868089814409395438151766493491876}{391924934868665403776503701936105107461283} a + \frac{58706189717293401996267381496548770781089}{391924934868665403776503701936105107461283}$

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:  \( 6115144487.96 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.12.11.3$x^{12} - 208$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$