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Properties

Label 20.0.65914313877...1801.1
Degree $20$
Signature $[0, 10]$
Discriminant $19^{10}\cdot 401^{10}$
Root discriminant $87.29$
Ramified primes $19, 401$
Class number $309488$ (GRH)
Class group $[2, 2, 58, 1334]$ (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![656899101, -742650399, 1045945552, -824644431, 668852055, -400713435, 233803276, -110220879, 49311908, -18317397, 6315713, -1778799, 448049, -82773, 12005, -330, -95, 12, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 21*x^18 + 12*x^17 - 95*x^16 - 330*x^15 + 12005*x^14 - 82773*x^13 + 448049*x^12 - 1778799*x^11 + 6315713*x^10 - 18317397*x^9 + 49311908*x^8 - 110220879*x^7 + 233803276*x^6 - 400713435*x^5 + 668852055*x^4 - 824644431*x^3 + 1045945552*x^2 - 742650399*x + 656899101)
 
gp: K = bnfinit(x^20 - 6*x^19 + 21*x^18 + 12*x^17 - 95*x^16 - 330*x^15 + 12005*x^14 - 82773*x^13 + 448049*x^12 - 1778799*x^11 + 6315713*x^10 - 18317397*x^9 + 49311908*x^8 - 110220879*x^7 + 233803276*x^6 - 400713435*x^5 + 668852055*x^4 - 824644431*x^3 + 1045945552*x^2 - 742650399*x + 656899101, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 21 x^{18} + 12 x^{17} - 95 x^{16} - 330 x^{15} + 12005 x^{14} - 82773 x^{13} + 448049 x^{12} - 1778799 x^{11} + 6315713 x^{10} - 18317397 x^{9} + 49311908 x^{8} - 110220879 x^{7} + 233803276 x^{6} - 400713435 x^{5} + 668852055 x^{4} - 824644431 x^{3} + 1045945552 x^{2} - 742650399 x + 656899101 \)

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:  \(659143138772292919540205490019944661801=19^{10}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.29$
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:  $19, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{15} - \frac{1}{9} a^{13} - \frac{4}{27} a^{12} - \frac{1}{27} a^{11} + \frac{4}{27} a^{10} + \frac{4}{9} a^{8} + \frac{8}{27} a^{7} + \frac{2}{9} a^{6} - \frac{2}{9} a^{5} - \frac{2}{27} a^{4} + \frac{1}{27} a^{3} + \frac{2}{27} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{27} a^{16} + \frac{2}{27} a^{13} + \frac{2}{27} a^{12} - \frac{2}{27} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{27} a^{8} - \frac{1}{9} a^{7} + \frac{1}{27} a^{5} - \frac{11}{27} a^{4} - \frac{1}{27} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{17} + \frac{1}{81} a^{16} - \frac{1}{81} a^{15} - \frac{1}{81} a^{14} + \frac{1}{81} a^{13} - \frac{8}{81} a^{12} - \frac{10}{81} a^{11} - \frac{10}{81} a^{10} - \frac{7}{81} a^{9} + \frac{20}{81} a^{8} - \frac{29}{81} a^{7} - \frac{11}{81} a^{6} + \frac{20}{81} a^{5} + \frac{38}{81} a^{4} + \frac{7}{81} a^{3} + \frac{25}{81} a^{2} - \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{2511} a^{18} + \frac{5}{2511} a^{17} - \frac{2}{279} a^{16} - \frac{5}{2511} a^{15} - \frac{13}{837} a^{14} - \frac{10}{2511} a^{13} + \frac{41}{837} a^{12} - \frac{314}{2511} a^{11} - \frac{254}{2511} a^{10} + \frac{415}{2511} a^{9} - \frac{34}{279} a^{8} + \frac{287}{2511} a^{7} + \frac{211}{837} a^{6} - \frac{1019}{2511} a^{5} + \frac{43}{837} a^{4} + \frac{893}{2511} a^{3} - \frac{854}{2511} a^{2} - \frac{83}{279} a + \frac{64}{279}$, $\frac{1}{1172744290403517368164827216110580473737079928624707212698141} a^{19} - \frac{51400287977168833819687899400071035132416975164561645560}{1172744290403517368164827216110580473737079928624707212698141} a^{18} + \frac{4824317835324817003144338940431074447801470975985175171664}{1172744290403517368164827216110580473737079928624707212698141} a^{17} + \frac{6203157371173229990236758834661620116864826588577829823328}{1172744290403517368164827216110580473737079928624707212698141} a^{16} - \frac{16186428513980920062023136333093182814570074960964644083972}{1172744290403517368164827216110580473737079928624707212698141} a^{15} + \frac{34326239649019529048462726037367907498254196841181658053148}{1172744290403517368164827216110580473737079928624707212698141} a^{14} - \frac{178734076788242331489806851112884161110095588727552197234358}{1172744290403517368164827216110580473737079928624707212698141} a^{13} + \frac{62740051308982023089484359142911647201293917209494280885678}{1172744290403517368164827216110580473737079928624707212698141} a^{12} + \frac{35443886141321649866698279686875206124975126665729045895936}{390914763467839122721609072036860157912359976208235737566047} a^{11} - \frac{15642486535260702930308699768884310450742756090740013804179}{390914763467839122721609072036860157912359976208235737566047} a^{10} + \frac{151270018836562830843855049856302563004592263706203280122895}{1172744290403517368164827216110580473737079928624707212698141} a^{9} + \frac{583833108566415587134243514098601328632363250843447628492895}{1172744290403517368164827216110580473737079928624707212698141} a^{8} + \frac{365360242786100055563782280510604323538021357282941415062734}{1172744290403517368164827216110580473737079928624707212698141} a^{7} - \frac{42142068645399485630085515035916295680490115501186649395847}{1172744290403517368164827216110580473737079928624707212698141} a^{6} + \frac{166839464713056836252198055387077433876956170227701285161754}{1172744290403517368164827216110580473737079928624707212698141} a^{5} - \frac{65616459153247405607032321787737182198806618069122491417556}{1172744290403517368164827216110580473737079928624707212698141} a^{4} - \frac{388992447607146545098981975615178502313755021682429868256922}{1172744290403517368164827216110580473737079928624707212698141} a^{3} - \frac{454204373549635901915083676427515142030178246832955504925363}{1172744290403517368164827216110580473737079928624707212698141} a^{2} + \frac{3296581536023802981783887474181844601753372128975026355820}{43434973718648791413512119115206684212484441800915081951783} a - \frac{485741900991793202908503133627170566444569534812097429805}{1569938809107787641452245269224337983583775004852352359703}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{58}\times C_{1334}$, which has order $309488$ (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:  \( -1 \) (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:  \( 795087.603907 \) (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

\(\Q(\sqrt{401}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-7619}) \), \(\Q(\sqrt{-19}, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.64024396763274499.1 x5, 10.0.25673783102073074099.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed