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Properties

Label 20.0.43245808601...3824.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 727^{10}$
Root discriminant $53.93$
Ramified primes $2, 727$
Class number $1573$ (GRH)
Class group $[11, 143]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, 0, 910336, 0, 499776, 0, -319648, 0, 159924, 0, 1048, 0, 17713, 0, -936, 0, 26, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 16*x^18 + 26*x^16 - 936*x^14 + 17713*x^12 + 1048*x^10 + 159924*x^8 - 319648*x^6 + 499776*x^4 + 910336*x^2 + 1048576)
 
gp: K = bnfinit(x^20 - 16*x^18 + 26*x^16 - 936*x^14 + 17713*x^12 + 1048*x^10 + 159924*x^8 - 319648*x^6 + 499776*x^4 + 910336*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{20} - 16 x^{18} + 26 x^{16} - 936 x^{14} + 17713 x^{12} + 1048 x^{10} + 159924 x^{8} - 319648 x^{6} + 499776 x^{4} + 910336 x^{2} + 1048576 \)

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:  \(43245808601363782342974430143053824=2^{20}\cdot 727^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.93$
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:  $2, 727$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{80} a^{12} + \frac{1}{80} a^{10} + \frac{3}{80} a^{8} - \frac{9}{80} a^{6} - \frac{1}{4} a^{5} - \frac{1}{5} a^{4} + \frac{1}{4} a^{3} + \frac{3}{20} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{160} a^{13} + \frac{1}{160} a^{11} + \frac{3}{160} a^{9} + \frac{11}{160} a^{7} - \frac{9}{40} a^{5} - \frac{7}{40} a^{3} + \frac{3}{10} a$, $\frac{1}{320} a^{14} - \frac{1}{320} a^{13} + \frac{1}{320} a^{12} - \frac{1}{320} a^{11} + \frac{3}{320} a^{10} - \frac{3}{320} a^{9} + \frac{11}{320} a^{8} - \frac{11}{320} a^{7} - \frac{9}{80} a^{6} - \frac{11}{80} a^{5} + \frac{13}{80} a^{4} - \frac{13}{80} a^{3} - \frac{1}{10} a^{2} + \frac{7}{20} a$, $\frac{1}{320} a^{15} + \frac{1}{160} a^{11} + \frac{1}{40} a^{9} + \frac{33}{320} a^{7} - \frac{9}{40} a^{5} - \frac{1}{4} a^{4} + \frac{19}{80} a^{3} - \frac{1}{4} a^{2} + \frac{7}{20} a$, $\frac{1}{8960} a^{16} - \frac{1}{320} a^{13} - \frac{17}{4480} a^{12} - \frac{1}{320} a^{11} + \frac{43}{2240} a^{10} - \frac{3}{320} a^{9} + \frac{15}{256} a^{8} - \frac{11}{320} a^{7} - \frac{127}{2240} a^{6} - \frac{11}{80} a^{5} + \frac{373}{2240} a^{4} + \frac{27}{80} a^{3} - \frac{43}{112} a^{2} - \frac{3}{20} a + \frac{2}{35}$, $\frac{1}{71680} a^{17} - \frac{3}{2560} a^{15} + \frac{53}{35840} a^{13} - \frac{1}{160} a^{12} - \frac{61}{4480} a^{11} - \frac{1}{160} a^{10} + \frac{199}{10240} a^{9} - \frac{3}{160} a^{8} + \frac{193}{3584} a^{7} - \frac{11}{160} a^{6} - \frac{379}{3584} a^{5} + \frac{9}{40} a^{4} - \frac{579}{4480} a^{3} - \frac{13}{40} a^{2} - \frac{61}{280} a + \frac{1}{5}$, $\frac{1}{19556038301104190771200} a^{18} + \frac{27726812234505619}{977801915055209538560} a^{16} - \frac{4999090263468542027}{9778019150552095385600} a^{14} - \frac{1}{320} a^{13} + \frac{32534104682716043}{34921496966257483520} a^{12} - \frac{1}{320} a^{11} + \frac{356503021852382031793}{19556038301104190771200} a^{10} - \frac{3}{320} a^{9} + \frac{220595731071922245129}{4889009575276047692800} a^{8} + \frac{29}{320} a^{7} - \frac{28054040572259426763}{977801915055209538560} a^{6} - \frac{11}{80} a^{5} - \frac{29575306118535996089}{174607484831287417600} a^{4} + \frac{37}{80} a^{3} + \frac{3231219831626170991}{9548846826711030650} a^{2} + \frac{1}{10} a + \frac{1745110179230090004}{4774423413355515325}$, $\frac{1}{156448306408833526169600} a^{19} + \frac{1388727280053}{24445047876380238464} a^{17} + \frac{109587071657063825773}{78224153204416763084800} a^{15} - \frac{11685199214117428653}{3911207660220838154240} a^{13} - \frac{1}{160} a^{12} - \frac{1616561556741165787087}{156448306408833526169600} a^{11} - \frac{1}{160} a^{10} + \frac{343767170449045821957}{19556038301104190771200} a^{9} - \frac{3}{160} a^{8} + \frac{8012134454402456799}{1117487902920239472640} a^{7} - \frac{11}{160} a^{6} + \frac{782005559665047663181}{4889009575276047692800} a^{5} + \frac{9}{40} a^{4} - \frac{864171973128838819723}{2444504787638023846400} a^{3} - \frac{13}{40} a^{2} - \frac{13344961814221690939}{152781549227376490400} a + \frac{1}{5}$

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

Class group and class number

$C_{11}\times C_{143}$, which has order $1573$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{11753995129}{15479302717276160} a^{19} + \frac{12025706041}{967456419829760} a^{17} - \frac{34724623265}{1547930271727616} a^{15} + \frac{266272007209}{386982567931904} a^{13} - \frac{211201525301033}{15479302717276160} a^{11} + \frac{931556113617}{386982567931904} a^{9} - \frac{342909078305733}{3869825679319040} a^{7} + \frac{130469015565981}{483728209914880} a^{5} - \frac{5716334234949}{48372820991488} a^{3} - \frac{13685749653241}{15116506559840} a \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1212427478.64 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{727}) \), \(\Q(\sqrt{-727}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{727})\), 5.5.8456464.1 x5, 10.10.207956266078624768.1, 10.0.51989066519656192.1 x5, 10.0.286047133533184.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
727Data not computed