↑

Properties

Label 21.21.3513559376...1472.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{35}\cdot 29^{12}$
Root discriminant $317.82$
Ramified primes $2, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:D_7:C_3$ (as 21T19)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6727656872, -146449603860, -345735927544, -168971552743, 168986405644, 138562884152, -31779062112, -39262594646, 2942499210, 5859384083, -143018778, -517708667, 3493224, 28285005, -33706, -959637, 0, 19544, 0, -217, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872)
 
gp: K = bnfinit(x^21 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872, 1)
 

Normalized defining polynomial

\( x^{21} - 217 x^{19} + 19544 x^{17} - 959637 x^{15} - 33706 x^{14} + 28285005 x^{13} + 3493224 x^{12} - 517708667 x^{11} - 143018778 x^{10} + 5859384083 x^{9} + 2942499210 x^{8} - 39262594646 x^{7} - 31779062112 x^{6} + 138562884152 x^{5} + 168986405644 x^{4} - 168971552743 x^{3} - 345735927544 x^{2} - 146449603860 x - 6727656872 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35135593769377589230177713282906868218673215743721472=2^{18}\cdot 7^{35}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $317.82$
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:  $2, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{203} a^{11} + \frac{1}{7} a^{10} + \frac{73}{203} a^{9} - \frac{3}{7} a^{8} - \frac{89}{203} a^{7} - \frac{8}{29} a^{5} + \frac{79}{203} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{609} a^{12} + \frac{1}{609} a^{11} - \frac{130}{609} a^{10} - \frac{304}{609} a^{9} + \frac{38}{203} a^{8} + \frac{8}{87} a^{7} - \frac{37}{87} a^{6} + \frac{23}{609} a^{5} + \frac{79}{609} a^{4} + \frac{2}{7} a^{3} + \frac{8}{21} a^{2} - \frac{1}{7} a + \frac{1}{3}$, $\frac{1}{1218} a^{13} + \frac{1}{1218} a^{11} - \frac{299}{1218} a^{9} - \frac{10}{21} a^{8} + \frac{39}{406} a^{7} + \frac{47}{203} a^{6} + \frac{83}{174} a^{5} + \frac{85}{609} a^{4} - \frac{1}{6} a^{3} + \frac{2}{21} a^{2} - \frac{5}{42} a - \frac{2}{21}$, $\frac{1}{35322} a^{14} + \frac{5}{11774} a^{12} - \frac{1}{609} a^{11} + \frac{1487}{5046} a^{10} + \frac{82}{203} a^{9} + \frac{3471}{11774} a^{8} - \frac{4180}{17661} a^{7} - \frac{15}{58} a^{6} - \frac{100}{203} a^{5} + \frac{15}{406} a^{4} + \frac{1}{3} a^{3} - \frac{17}{42} a^{2} - \frac{10}{21} a - \frac{4}{21}$, $\frac{1}{35322} a^{15} - \frac{1}{2523} a^{13} - \frac{1}{17661} a^{11} - \frac{8}{21} a^{10} + \frac{2736}{5887} a^{9} + \frac{830}{5887} a^{8} + \frac{26}{609} a^{7} - \frac{13}{87} a^{6} + \frac{13}{87} a^{5} - \frac{16}{609} a^{4} - \frac{2}{21} a^{3} + \frac{8}{21} a^{2} + \frac{3}{14} a + \frac{1}{7}$, $\frac{1}{35322} a^{16} - \frac{4}{5887} a^{12} - \frac{1}{609} a^{11} + \frac{5323}{17661} a^{10} - \frac{6355}{17661} a^{9} - \frac{2654}{17661} a^{8} - \frac{3884}{17661} a^{7} + \frac{20}{87} a^{6} + \frac{6}{203} a^{5} + \frac{124}{609} a^{4} - \frac{8}{21} a^{3} + \frac{1}{6} a^{2} - \frac{2}{21} a + \frac{3}{7}$, $\frac{1}{1024338} a^{17} - \frac{1}{73167} a^{15} - \frac{59}{512169} a^{13} + \frac{610}{512169} a^{11} + \frac{56314}{512169} a^{10} + \frac{2582}{17661} a^{9} + \frac{286}{17661} a^{8} - \frac{10}{29} a^{7} - \frac{181}{609} a^{6} + \frac{1}{29} a^{5} + \frac{128}{609} a^{4} - \frac{449}{1218} a^{3} - \frac{3}{7} a^{2} + \frac{5}{21} a$, $\frac{1}{1024338} a^{18} - \frac{1}{73167} a^{16} - \frac{1}{512169} a^{14} - \frac{202}{512169} a^{12} + \frac{808}{512169} a^{11} - \frac{1859}{17661} a^{10} + \frac{7739}{17661} a^{9} + \frac{185}{5887} a^{8} + \frac{6016}{17661} a^{7} - \frac{13}{87} a^{6} - \frac{194}{609} a^{5} - \frac{107}{406} a^{4} - \frac{8}{21} a^{3} - \frac{2}{7} a^{2} - \frac{10}{21} a + \frac{1}{7}$, $\frac{1}{29705802} a^{19} - \frac{1}{2121843} a^{17} - \frac{59}{14852901} a^{15} + \frac{3974}{14852901} a^{13} + \frac{2512}{4950967} a^{12} - \frac{724}{512169} a^{11} - \frac{55691}{170723} a^{10} + \frac{5147}{17661} a^{9} - \frac{848}{17661} a^{8} + \frac{47}{609} a^{7} - \frac{1679}{5887} a^{6} - \frac{10019}{35322} a^{5} - \frac{139}{609} a^{4} - \frac{176}{609} a^{3} + \frac{4}{21} a^{2} - \frac{4}{21} a - \frac{4}{21}$, $\frac{1}{17792022752530611079532327640320616811149720030413412} a^{20} + \frac{18694682276312134497982345768977417346181113}{4448005688132652769883081910080154202787430007603353} a^{19} - \frac{2284698150384745391002182772718456815400832887}{17792022752530611079532327640320616811149720030413412} a^{18} - \frac{1182189242560141238616839850570352608417210909}{2965337125421768513255387940053436135191620005068902} a^{17} + \frac{8238352223492449746226786851933251872089441586}{635429384018950395697583130011450600398204286800479} a^{16} - \frac{123640784580338474738458102436973394419038018065}{8896011376265305539766163820160308405574860015206706} a^{15} - \frac{21043797136030981164334943042543865373237407877}{2541717536075801582790332520045802401592817147201916} a^{14} - \frac{32801175559589820051214569020977685786713182468}{211809794672983465232527710003816866799401428933493} a^{13} - \frac{1699870557570381669574324300556237551212633373055}{2541717536075801582790332520045802401592817147201916} a^{12} - \frac{192876393391526926829214669302676040239005027323}{102253004324888569422599584139773659834193793278238} a^{11} - \frac{31990502688820529398664140497395567765602560785415}{204506008649777138845199168279547319668387586556476} a^{10} + \frac{1436464234858423540257435105067821773506420192625}{5288948499563201866686185386540016887975541031633} a^{9} + \frac{2467895691389645342337464760465444286997250863435}{7051931332750935822248247182053355850634054708844} a^{8} + \frac{1009914710488624855750113794371100606836124656541}{5288948499563201866686185386540016887975541031633} a^{7} - \frac{1072824633615086013005429482434459995007045118789}{5288948499563201866686185386540016887975541031633} a^{6} + \frac{1329160327324982977578012586151331983100942473539}{3525965666375467911124123591026677925317027354422} a^{5} + \frac{80921748625903260284930966720959794715194032842}{182377534467696616092627082294483340964673828677} a^{4} - \frac{3223074142113466190718542150122478436559598450}{60792511489232205364209027431494446988224609559} a^{3} - \frac{5013336199939937684236068629099758870322225897}{25155521995544360840362356178549426339955010852} a^{2} - \frac{2385937425975839349892587824532003137870387720}{6288880498886090210090589044637356584988752713} a - \frac{393817543068785918236506664766566774673002769}{6288880498886090210090589044637356584988752713}$

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:  $20$
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:  \( 100927220322000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:D_7:C_3$ (as 21T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 294
The 14 conjugacy class representatives for $C_7:D_7:C_3$
Character table for $C_7:D_7:C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 21 sibling: data not computed
Degree 42 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed
$29$29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.4$x^{7} - 1856$$7$$1$$6$$C_7$$[\ ]_{7}$