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Properties

Label 21.5.14555496170...2301.1
Degree $21$
Signature $[5, 8]$
Discriminant $3^{22}\cdot 19^{12}\cdot 271^{12}\cdot 2377^{12}\cdot 8689^{12}\cdot 280909^{3}$
Root discriminant $37{,}948{,}250.84$
Ramified primes $3, 19, 271, 2377, 8689, 280909$
Class number Not computed
Class group Not computed
Galois group 21T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29286779250664128412587474959348265487749178557406171188181646773603899348452864889344129965020289, 0, 0, -3119377615169669450854789652811784887721330210462390605953766388930476432412202071090, 0, 0, -20154731914920171427946883694168074743974027040776144815703995172616607, 0, 0, 2274824288784193137086018848627059640970063875355389053578, 0, 0, -31735648244136873749575014987723762737667447, 0, 0, 209205884459568917365972820312, 0, 0, -747855216958107, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 747855216958107*x^18 + 209205884459568917365972820312*x^15 - 31735648244136873749575014987723762737667447*x^12 + 2274824288784193137086018848627059640970063875355389053578*x^9 - 20154731914920171427946883694168074743974027040776144815703995172616607*x^6 - 3119377615169669450854789652811784887721330210462390605953766388930476432412202071090*x^3 - 29286779250664128412587474959348265487749178557406171188181646773603899348452864889344129965020289)
 
gp: K = bnfinit(x^21 - 747855216958107*x^18 + 209205884459568917365972820312*x^15 - 31735648244136873749575014987723762737667447*x^12 + 2274824288784193137086018848627059640970063875355389053578*x^9 - 20154731914920171427946883694168074743974027040776144815703995172616607*x^6 - 3119377615169669450854789652811784887721330210462390605953766388930476432412202071090*x^3 - 29286779250664128412587474959348265487749178557406171188181646773603899348452864889344129965020289, 1)
 

Normalized defining polynomial

\( x^{21} - 747855216958107 x^{18} + 209205884459568917365972820312 x^{15} - 31735648244136873749575014987723762737667447 x^{12} + 2274824288784193137086018848627059640970063875355389053578 x^{9} - 20154731914920171427946883694168074743974027040776144815703995172616607 x^{6} - 3119377615169669450854789652811784887721330210462390605953766388930476432412202071090 x^{3} - 29286779250664128412587474959348265487749178557406171188181646773603899348452864889344129965020289 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1455549617007968751393125676744930377861970587733391324997627304594771456558262344297508842187835376784067562538854669999379967568737747192168449322823799142301=3^{22}\cdot 19^{12}\cdot 271^{12}\cdot 2377^{12}\cdot 8689^{12}\cdot 280909^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37{,}948{,}250.84$
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:  $3, 19, 271, 2377, 8689, 280909$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{165091} a^{5} - \frac{23787}{165091} a^{2}$, $\frac{1}{17556796244356927} a^{6} - \frac{747855216958107}{17556796244356927} a^{3}$, $\frac{1}{17556796244356927} a^{7} - \frac{747855216958107}{17556796244356927} a^{4}$, $\frac{1}{2898469048777129435357} a^{8} - \frac{747855216958107}{2898469048777129435357} a^{5} - \frac{2568}{165091} a^{2}$, $\frac{1}{308241094365865496762091772883329} a^{9} - \frac{747855216958107}{308241094365865496762091772883329} a^{6} + \frac{11915948761256}{17556796244356927} a^{3}$, $\frac{1}{308241094365865496762091772883329} a^{10} - \frac{747855216958107}{308241094365865496762091772883329} a^{7} + \frac{11915948761256}{17556796244356927} a^{4}$, $\frac{1}{152663491529865302177851478631245003817} a^{11} + \frac{1}{924723283097596490286275318649987} a^{10} + \frac{1}{924723283097596490286275318649987} a^{9} - \frac{18304651461315034}{152663491529865302177851478631245003817} a^{8} - \frac{18304651461315034}{924723283097596490286275318649987} a^{7} - \frac{18304651461315034}{924723283097596490286275318649987} a^{6} - \frac{16797025078637564}{8695407146331388306071} a^{5} - \frac{16797025078637564}{52670388733070781} a^{4} - \frac{16797025078637564}{52670388733070781} a^{3} - \frac{179330}{495273} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{16235178263757289453351836821712404302908611909949} a^{12} + \frac{16808941027398820}{16235178263757289453351836821712404302908611909949} a^{9} - \frac{735939268196851}{924723283097596490286275318649987} a^{6} + \frac{5856203078630480}{17556796244356927} a^{3} - \frac{1}{3}$, $\frac{1}{16235178263757289453351836821712404302908611909949} a^{13} + \frac{16808941027398820}{16235178263757289453351836821712404302908611909949} a^{10} - \frac{735939268196851}{924723283097596490286275318649987} a^{7} + \frac{5856203078630480}{17556796244356927} a^{4} - \frac{1}{3} a$, $\frac{1}{2680281814741954673143308092733322538771485648825390359} a^{14} - \frac{249285072319369}{893427271580651557714436030911107512923828549608463453} a^{11} - \frac{1}{924723283097596490286275318649987} a^{10} - \frac{1}{924723283097596490286275318649987} a^{9} + \frac{5856237397706061}{50887830509955100725950492877081667939} a^{8} + \frac{18304651461315034}{924723283097596490286275318649987} a^{7} + \frac{18304651461315034}{924723283097596490286275318649987} a^{6} - \frac{18304754418541777}{8695407146331388306071} a^{5} + \frac{16797025078637564}{52670388733070781} a^{4} + \frac{16797025078637564}{52670388733070781} a^{3} + \frac{215590}{495273} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{285037716767599194274376439974657748574778699186470531554738366723} a^{15} - \frac{249285072319369}{95012572255866398091458813324885916191592899728823510518246122241} a^{12} - \frac{17544880295595671}{16235178263757289453351836821712404302908611909949} a^{9} + \frac{18304548504088291}{924723283097596490286275318649987} a^{6} - \frac{18316566989725324}{52670388733070781} a^{3} + \frac{1}{3}$, $\frac{1}{285037716767599194274376439974657748574778699186470531554738366723} a^{16} - \frac{249285072319369}{95012572255866398091458813324885916191592899728823510518246122241} a^{13} - \frac{17544880295595671}{16235178263757289453351836821712404302908611909949} a^{10} + \frac{18304548504088291}{924723283097596490286275318649987} a^{7} - \frac{18316566989725324}{52670388733070781} a^{4} + \frac{1}{3} a$, $\frac{1}{47057161698879718581951080851856222369958790227393606524903311700666793} a^{17} - \frac{249285072319369}{15685720566293239527317026950618740789986263409131202174967770566888931} a^{14} + \frac{11915948761256}{2680281814741954673143308092733322538771485648825390359} a^{11} + \frac{1}{924723283097596490286275318649987} a^{10} + \frac{1}{924723283097596490286275318649987} a^{9} - \frac{34319075581}{50887830509955100725950492877081667939} a^{8} - \frac{18304651461315034}{924723283097596490286275318649987} a^{7} - \frac{18304651461315034}{924723283097596490286275318649987} a^{6} + \frac{5852265554902631}{2898469048777129435357} a^{5} - \frac{16797025078637564}{52670388733070781} a^{4} - \frac{16797025078637564}{52670388733070781} a^{3} + \frac{94268}{495273} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{182766477808092522399709596436633816323141439818105534822986238946749896141430058888284125510088540412256423803964359848018402604869} a^{18} - \frac{1}{141171485096639155745853242555568667109876370682180819574709935102000379} a^{17} - \frac{168648867581890736767111998526285930232887469227852638421190179169}{182766477808092522399709596436633816323141439818105534822986238946749896141430058888284125510088540412256423803964359848018402604869} a^{15} + \frac{249285072319369}{47057161698879718581951080851856222369958790227393606524903311700666793} a^{14} + \frac{1}{48705534791271868360055510465137212908725835729847} a^{13} - \frac{32623268843064629075100813997510680986131356845921975915669657729}{3470004345977738504990975611744628246701999583781498356887143210588748537343007589115118864674800864904713493828649} a^{12} + \frac{17544880295595671}{8040845444225864019429924278199967616314456946476171077} a^{11} - \frac{35861447705671961}{48705534791271868360055510465137212908725835729847} a^{10} + \frac{337556247663447470793787911714018590257982974918006981789249581271}{592933522326385866989135452820548982976345274973261065813224031360965027224328457216061242043064261} a^{9} - \frac{18304548504088291}{457990474589595906533554435893735011451} a^{8} - \frac{51162762350393311}{2774169849292789470858825955949961} a^{7} + \frac{645615647745764583098855104218501818784088379223974386921484966566}{33772307548250180545460407897697273974619801372533485555710694386869474693033653243} a^{6} + \frac{18316566989725324}{26086221438994164918213} a^{5} + \frac{24148938591184258}{52670388733070781} a^{4} - \frac{171898102908185026325045568186072696563229253236156793624165474646}{641201030799401745803313918418418854470166556224403491848338298903} a^{3} + \frac{645587}{1485819} a^{2} + \frac{2}{9} a - \frac{54742793395740176982120344470809029635418155290541}{109564584883559618009610675953228329422226547583867}$, $\frac{1}{182766477808092522399709596436633816323141439818105534822986238946749896141430058888284125510088540412256423803964359848018402604869} a^{19} + \frac{1}{141171485096639155745853242555568667109876370682180819574709935102000379} a^{17} - \frac{168648867581890736767111998526285930232887469227852638421190179169}{182766477808092522399709596436633816323141439818105534822986238946749896141430058888284125510088540412256423803964359848018402604869} a^{16} + \frac{1}{855113150302797582823129319923973245724336097559411594664215100169} a^{15} + \frac{16808941027398820}{141171485096639155745853242555568667109876370682180819574709935102000379} a^{14} - \frac{311603483462327802493073748132004994448449589279233758363121739488}{10410013037933215514972926835233884740105998751344495070661429631766245612029022767345356594024402594714140481485947} a^{13} - \frac{18304651461315034}{855113150302797582823129319923973245724336097559411594664215100169} a^{12} - \frac{735939268196851}{8040845444225864019429924278199967616314456946476171077} a^{11} - \frac{508274188246974922643465364280179700801917086163196746872289477292}{592933522326385866989135452820548982976345274973261065813224031360965027224328457216061242043064261} a^{10} + \frac{6105522470025430}{16235178263757289453351836821712404302908611909949} a^{9} + \frac{7804333107662469}{50887830509955100725950492877081667939} a^{8} + \frac{227295890715868077874608320051002044501089444671701017839991728934}{11257435849416726848486802632565757991539933790844495185236898128956491564344551081} a^{7} + \frac{69467310854481602}{2774169849292789470858825955949961} a^{6} - \frac{2991523404708205}{26086221438994164918213} a^{5} - \frac{29601984339883685074495860595942341707974331543128536317133010913}{641201030799401745803313918418418854470166556224403491848338298903} a^{4} + \frac{67247783435934245}{158011166199212343} a^{3} + \frac{100718}{495273} a^{2} - \frac{42568950630900219425496936031561437477392983336778}{109564584883559618009610675953228329422226547583867} a - \frac{1}{9}$, $\frac{1}{30173100587815802615490456985320313370603743441010860849461621173957887103884829851925714564586027225199825262220280131669206104440428079} a^{20} + \frac{258818486284377093768430613752659972747223568255083022811035353433}{30173100587815802615490456985320313370603743441010860849461621173957887103884829851925714564586027225199825262220280131669206104440428079} a^{17} + \frac{1}{855113150302797582823129319923973245724336097559411594664215100169} a^{16} + \frac{311389003692847408042576273159026830221072939175834545879392546133}{1718599462445432482582395464155597265628839439858214035710566079337921254335483397683812265464082648763952166228996476177} a^{14} - \frac{249285072319369}{285037716767599194274376439974657748574778699186470531554738366723} a^{13} - \frac{1}{48705534791271868360055510465137212908725835729847} a^{12} + \frac{106545943264149258515613297075723683407185049547918403803247060215}{32629329378128456389034453680532417382849272596870214205390322853804359103163869776752255503377173970917} a^{11} + \frac{35125508437475110}{48705534791271868360055510465137212908725835729847} a^{10} - \frac{69479329760469601}{48705534791271868360055510465137212908725835729847} a^{9} + \frac{432169589815677761336723147174520074830076800804958083248681645103}{5575504025448170556430604200238740657743957628392925663877834247022668446547618847540113} a^{8} - \frac{36609405879856811}{2774169849292789470858825955949961} a^{7} + \frac{2979504919071172}{2774169849292789470858825955949961} a^{6} - \frac{506592533438048425591207294888273985716982653396623311191821814999}{317569558127112100849244694316845561310002800800928990618202054312585519} a^{5} - \frac{68707642225638016}{158011166199212343} a^{4} + \frac{3896225738988397}{17556796244356927} a^{3} + \frac{5737988653671799534477008460227367773918265180875269405}{18088126883011740896824636103794418132644802967168186897} a^{2} + \frac{4}{9} a + \frac{1}{9}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $12$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7348320
The 118 conjugacy class representatives for t21n138 are not computed
Character table for t21n138 is not computed

Intermediate fields

7.5.7584543.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ $15{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$

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
3Data not computed
19Data not computed
271Data not computed
2377Data not computed
8689Data not computed
280909Data not computed