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Properties

Label 20.20.3547256202...9744.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{15}\cdot 19^{11}\cdot 43^{11}$
Root discriminant $67.22$
Ramified primes $2, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83, 1528, -20724, 36702, 121081, -437006, 321800, 353964, -605673, 91048, 283437, -139914, -42369, 41062, -1026, -4834, 698, 238, -49, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 49*x^18 + 238*x^17 + 698*x^16 - 4834*x^15 - 1026*x^14 + 41062*x^13 - 42369*x^12 - 139914*x^11 + 283437*x^10 + 91048*x^9 - 605673*x^8 + 353964*x^7 + 321800*x^6 - 437006*x^5 + 121081*x^4 + 36702*x^3 - 20724*x^2 + 1528*x + 83)
 
gp: K = bnfinit(x^20 - 4*x^19 - 49*x^18 + 238*x^17 + 698*x^16 - 4834*x^15 - 1026*x^14 + 41062*x^13 - 42369*x^12 - 139914*x^11 + 283437*x^10 + 91048*x^9 - 605673*x^8 + 353964*x^7 + 321800*x^6 - 437006*x^5 + 121081*x^4 + 36702*x^3 - 20724*x^2 + 1528*x + 83, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 49 x^{18} + 238 x^{17} + 698 x^{16} - 4834 x^{15} - 1026 x^{14} + 41062 x^{13} - 42369 x^{12} - 139914 x^{11} + 283437 x^{10} + 91048 x^{9} - 605673 x^{8} + 353964 x^{7} + 321800 x^{6} - 437006 x^{5} + 121081 x^{4} + 36702 x^{3} - 20724 x^{2} + 1528 x + 83 \)

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:  \(3547256202968983472963275805780639744=2^{15}\cdot 19^{11}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.22$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{190} a^{18} - \frac{33}{190} a^{17} - \frac{1}{19} a^{16} + \frac{21}{95} a^{15} - \frac{8}{19} a^{14} + \frac{13}{38} a^{13} + \frac{39}{190} a^{12} + \frac{21}{190} a^{11} + \frac{7}{19} a^{10} + \frac{44}{95} a^{9} - \frac{7}{19} a^{8} - \frac{28}{95} a^{7} - \frac{47}{95} a^{6} + \frac{73}{190} a^{5} + \frac{4}{19} a^{4} - \frac{13}{38} a^{3} + \frac{81}{190} a^{2} + \frac{34}{95} a + \frac{18}{95}$, $\frac{1}{1635049207473016373135289498792699850} a^{19} + \frac{1171481358494037393774671843297783}{1635049207473016373135289498792699850} a^{18} + \frac{213114943443319716843618314611399057}{1635049207473016373135289498792699850} a^{17} + \frac{171704226102436260838309792584496971}{817524603736508186567644749396349925} a^{16} - \frac{6228187541133996571444307578515924}{817524603736508186567644749396349925} a^{15} - \frac{1379205240923033714666679712491173}{3593514741698937083813823074269670} a^{14} + \frac{227122524628212307918100882125305669}{1635049207473016373135289498792699850} a^{13} + \frac{74717441951309611789547531359698934}{163504920747301637313528949879269985} a^{12} - \frac{111981267409237102395381185048223312}{817524603736508186567644749396349925} a^{11} - \frac{450824110715600152248409507023036817}{1635049207473016373135289498792699850} a^{10} + \frac{724413832375427287984519110592269083}{1635049207473016373135289498792699850} a^{9} + \frac{301262912667487614850931987195012049}{1635049207473016373135289498792699850} a^{8} - \frac{103822371554886367692460069946320297}{327009841494603274627057899758539970} a^{7} - \frac{307949228431371565840432439075313383}{817524603736508186567644749396349925} a^{6} - \frac{557774459939558228132378898196922657}{1635049207473016373135289498792699850} a^{5} - \frac{3245340862739495254945618011037703}{17211044289189646033003047355712630} a^{4} - \frac{31320093589232096049549748309794603}{125773015959462797933483807599438450} a^{3} - \frac{198236315390885168748947417112790691}{1635049207473016373135289498792699850} a^{2} + \frac{363071081189301730524975970908246372}{817524603736508186567644749396349925} a + \frac{19935121158223073927775694204962618}{817524603736508186567644749396349925}$

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:  \( 986564865952 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.3$x^{10} - 18 x^{8} + 216 x^{6} - 304 x^{4} + 656 x^{2} - 544$$2$$5$$15$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 3]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$