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Properties

Label 20.0.16537062841...8176.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 3^{16}\cdot 389^{4}$
Root discriminant $18.24$
Ramified primes $2, 3, 389$
Class number $2$
Class group $[2]$
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -4, -1, 18, 5, -104, 265, -380, 442, -448, 313, -112, 5, 12, -1, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^17 - x^16 + 12*x^15 + 5*x^14 - 112*x^13 + 313*x^12 - 448*x^11 + 442*x^10 - 380*x^9 + 265*x^8 - 104*x^7 + 5*x^6 + 18*x^5 - x^4 - 4*x^3 + 1)
 
gp: K = bnfinit(x^20 - 2*x^17 - x^16 + 12*x^15 + 5*x^14 - 112*x^13 + 313*x^12 - 448*x^11 + 442*x^10 - 380*x^9 + 265*x^8 - 104*x^7 + 5*x^6 + 18*x^5 - x^4 - 4*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{17} - x^{16} + 12 x^{15} + 5 x^{14} - 112 x^{13} + 313 x^{12} - 448 x^{11} + 442 x^{10} - 380 x^{9} + 265 x^{8} - 104 x^{7} + 5 x^{6} + 18 x^{5} - x^{4} - 4 x^{3} + 1 \)

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:  \(16537062841993940409778176=2^{24}\cdot 3^{16}\cdot 389^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.24$
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:  $2, 3, 389$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1621} a^{18} - \frac{325}{1621} a^{17} + \frac{94}{1621} a^{16} + \frac{704}{1621} a^{15} + \frac{366}{1621} a^{14} - \frac{757}{1621} a^{13} + \frac{480}{1621} a^{12} + \frac{349}{1621} a^{11} + \frac{107}{1621} a^{10} - \frac{760}{1621} a^{9} - \frac{502}{1621} a^{8} + \frac{392}{1621} a^{7} - \frac{36}{1621} a^{6} + \frac{17}{1621} a^{5} + \frac{456}{1621} a^{4} - \frac{251}{1621} a^{3} - \frac{606}{1621} a^{2} + \frac{325}{1621} a - \frac{166}{1621}$, $\frac{1}{15510447587730635} a^{19} + \frac{921189170403}{15510447587730635} a^{18} + \frac{1325655902828179}{15510447587730635} a^{17} - \frac{389341812093203}{3102089517546127} a^{16} - \frac{5827126722030526}{15510447587730635} a^{15} + \frac{5723870613502249}{15510447587730635} a^{14} - \frac{5806086327940168}{15510447587730635} a^{13} - \frac{4340190629231201}{15510447587730635} a^{12} + \frac{1109250139282375}{3102089517546127} a^{11} + \frac{650636017785012}{15510447587730635} a^{10} - \frac{3335484155148982}{15510447587730635} a^{9} - \frac{3206567893589451}{15510447587730635} a^{8} - \frac{7118982883008923}{15510447587730635} a^{7} - \frac{452366673795518}{15510447587730635} a^{6} + \frac{6378443607662316}{15510447587730635} a^{5} + \frac{101582771905588}{330009523143205} a^{4} + \frac{3176140929641592}{15510447587730635} a^{3} + \frac{1787650420293042}{15510447587730635} a^{2} - \frac{997695159117884}{15510447587730635} a - \frac{5107085228052042}{15510447587730635}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{1026454468164}{9568443915935} a^{19} + \frac{2672516630162}{9568443915935} a^{18} + \frac{1013241548436}{9568443915935} a^{17} - \frac{301129302918}{1913688783187} a^{16} - \frac{5946015002034}{9568443915935} a^{15} + \frac{7879253075676}{9568443915935} a^{14} + \frac{35027642440353}{9568443915935} a^{13} - \frac{91126858981824}{9568443915935} a^{12} + \frac{6428034651156}{1913688783187} a^{11} + \frac{270175022255118}{9568443915935} a^{10} - \frac{482657774185668}{9568443915935} a^{9} + \frac{461848428709326}{9568443915935} a^{8} - \frac{439573693820967}{9568443915935} a^{7} + \frac{351178980867738}{9568443915935} a^{6} - \frac{138517241402766}{9568443915935} a^{5} + \frac{235549335312}{203583913105} a^{4} + \frac{17234523083103}{9568443915935} a^{3} + \frac{24778925658543}{9568443915935} a^{2} - \frac{9998585763681}{9568443915935} a - \frac{1882240525848}{9568443915935} \) (order $4$)
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:  \( 81195.772932 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.0.112960521216.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
389Data not computed