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Properties

Label 21.3.426...647.2
Degree $21$
Signature $[3, 9]$
Discriminant $-4.260\times 10^{49}$
Root discriminant \(230.84\)
Ramified primes $7,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_7$ (as 21T4)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 2171*x^14 + 1219335*x^7 - 62748517)
 
Copy content gp:K = bnfinit(y^21 - 2171*y^14 + 1219335*y^7 - 62748517, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2171*x^14 + 1219335*x^7 - 62748517);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 2171*x^14 + 1219335*x^7 - 62748517)
 

\( x^{21} - 2171x^{14} + 1219335x^{7} - 62748517 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $21$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[3, 9]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-42600210199743647641790494093640979042918020345647\) \(\medspace = -\,7^{35}\cdot 13^{18}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(230.84\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $7^{71/42}13^{6/7}\approx 241.78352389457936$
Ramified primes:   \(7\), \(13\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$:   $C_3$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{26}a^{7}-\frac{1}{2}$, $\frac{1}{26}a^{8}-\frac{1}{2}a$, $\frac{1}{26}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{26}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{338}a^{11}-\frac{11}{26}a^{4}$, $\frac{1}{338}a^{12}-\frac{11}{26}a^{5}$, $\frac{1}{338}a^{13}-\frac{11}{26}a^{6}$, $\frac{1}{1467596}a^{14}+\frac{339}{56446}a^{7}-\frac{131}{668}$, $\frac{1}{1467596}a^{15}+\frac{339}{56446}a^{8}-\frac{131}{668}a$, $\frac{1}{19078748}a^{16}+\frac{339}{733798}a^{9}+\frac{1205}{8684}a^{2}$, $\frac{1}{19078748}a^{17}+\frac{339}{733798}a^{10}+\frac{1205}{8684}a^{3}$, $\frac{1}{248023724}a^{18}+\frac{339}{9539374}a^{11}+\frac{44625}{112892}a^{4}$, $\frac{1}{248023724}a^{19}+\frac{339}{9539374}a^{12}+\frac{44625}{112892}a^{5}$, $\frac{1}{3224308412}a^{20}-\frac{56107}{124011862}a^{13}+\frac{27257}{1467596}a^{6}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $11$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{3}{733798}a^{14}-\frac{137}{56446}a^{7}-\frac{280}{167}$, $\frac{11}{1467596}a^{14}-\frac{613}{56446}a^{7}+\frac{1231}{668}$, $\frac{12951}{1612154206}a^{20}-\frac{3345}{124011862}a^{19}+\frac{4747}{124011862}a^{18}+\frac{3977}{9539374}a^{17}-\frac{23837}{19078748}a^{16}-\frac{227}{733798}a^{15}+\frac{3097}{733798}a^{14}-\frac{1282434}{62005931}a^{13}+\frac{136080}{4769687}a^{12}+\frac{142159}{9539374}a^{11}-\frac{492793}{733798}a^{10}+\frac{686981}{366899}a^{9}-\frac{17133}{56446}a^{8}-\frac{273137}{56446}a^{7}+\frac{9700835}{733798}a^{6}+\frac{363379}{56446}a^{5}-\frac{1834720}{28223}a^{4}+\frac{471551}{2171}a^{3}-\frac{4152207}{8684}a^{2}+\frac{108806}{167}a-\frac{96558}{167}$, $\frac{58977}{3224308412}a^{20}-\frac{9983}{248023724}a^{19}-\frac{15211}{124011862}a^{18}-\frac{699}{19078748}a^{17}+\frac{605}{4769687}a^{16}+\frac{4925}{1467596}a^{15}-\frac{2173}{1467596}a^{14}-\frac{1814724}{62005931}a^{13}+\frac{341199}{9539374}a^{12}+\frac{1851055}{9539374}a^{11}+\frac{45269}{733798}a^{10}-\frac{280317}{733798}a^{9}-\frac{104170}{28223}a^{8}+\frac{49255}{56446}a^{7}+\frac{17395823}{1467596}a^{6}+\frac{1013853}{112892}a^{5}-\frac{1463005}{28223}a^{4}-\frac{43367}{8684}a^{3}+\frac{661293}{4342}a^{2}+\frac{69251}{668}a-\frac{259757}{668}$, $\frac{1178183}{3224308412}a^{20}+\frac{25256}{62005931}a^{19}-\frac{302775}{248023724}a^{18}-\frac{131693}{19078748}a^{17}-\frac{274167}{19078748}a^{16}+\frac{931}{733798}a^{15}+\frac{11935}{112892}a^{14}-\frac{56623197}{124011862}a^{13}-\frac{4785273}{9539374}a^{12}+\frac{7228251}{4769687}a^{11}+\frac{6326811}{733798}a^{10}+\frac{6602045}{366899}a^{9}-\frac{44777}{28223}a^{8}-\frac{573923}{4342}a^{7}+\frac{35585123}{1467596}a^{6}+\frac{1118869}{56446}a^{5}-\frac{8248441}{112892}a^{4}-\frac{3967237}{8684}a^{3}-\frac{8824425}{8684}a^{2}+\frac{21993}{334}a+\frac{4902383}{668}$, $\frac{15253343}{1612154206}a^{20}+\frac{386135}{62005931}a^{19}-\frac{442875}{9539374}a^{18}-\frac{1904569}{9539374}a^{17}-\frac{1532941}{4769687}a^{16}+\frac{315655}{733798}a^{15}+\frac{1447962}{366899}a^{14}-\frac{631680854}{62005931}a^{13}-\frac{36997371}{4769687}a^{12}+\frac{15566950}{366899}a^{11}+\frac{69691267}{366899}a^{10}+\frac{111853949}{366899}a^{9}-\frac{12345851}{28223}a^{8}-\frac{219527909}{56446}a^{7}+\frac{997024869}{733798}a^{6}+\frac{52561694}{28223}a^{5}+\frac{98131}{334}a^{4}-\frac{24426217}{4342}a^{3}-\frac{17837662}{2171}a^{2}+\frac{12726467}{334}a+\frac{78538437}{334}$, $\frac{11386185}{1612154206}a^{20}-\frac{4677380}{62005931}a^{19}+\frac{4153783}{124011862}a^{18}+\frac{1933895}{9539374}a^{17}-\frac{1405048}{4769687}a^{16}-\frac{177185}{366899}a^{15}+\frac{127163}{112892}a^{14}-\frac{942706526}{62005931}a^{13}+\frac{757095730}{4769687}a^{12}-\frac{346858372}{4769687}a^{11}-\frac{159094713}{366899}a^{10}+\frac{223486312}{366899}a^{9}+\frac{27798339}{28223}a^{8}-\frac{10748575}{4342}a^{7}+\frac{5933249471}{733798}a^{6}-\frac{2314378432}{28223}a^{5}+\frac{2206268609}{56446}a^{4}+\frac{993363519}{4342}a^{3}-\frac{667437310}{2171}a^{2}-\frac{81450170}{167}a+\frac{901978019}{668}$, $\frac{11012992}{806077103}a^{20}+\frac{3093841}{124011862}a^{19}-\frac{18030917}{124011862}a^{18}+\frac{351037}{9539374}a^{17}+\frac{8939855}{9539374}a^{16}-\frac{868175}{733798}a^{15}-\frac{8737945}{1467596}a^{14}-\frac{798663873}{62005931}a^{13}-\frac{141831602}{4769687}a^{12}+\frac{753638800}{4769687}a^{11}-\frac{25443771}{366899}a^{10}-\frac{315310474}{366899}a^{9}+\frac{30394290}{28223}a^{8}+\frac{166180855}{28223}a^{7}+\frac{101853493}{366899}a^{6}+\frac{353080883}{56446}a^{5}-\frac{1249284459}{56446}a^{4}+\frac{142098237}{4342}a^{3}-\frac{15612267}{4342}a^{2}+\frac{3328227}{334}a-\frac{251442859}{668}$, $\frac{199253793997745}{3224308412}a^{20}-\frac{53317207037799}{248023724}a^{19}-\frac{72932952075158}{62005931}a^{18}-\frac{46711085994827}{19078748}a^{17}+\frac{2252093894559}{4769687}a^{16}+\frac{30130796745689}{1467596}a^{15}+\frac{7351964918405}{112892}a^{14}-\frac{37\cdots 59}{62005931}a^{13}+\frac{20\cdots 59}{9539374}a^{12}+\frac{10\cdots 37}{9539374}a^{11}+\frac{878124963305723}{366899}a^{10}-\frac{329298680301219}{733798}a^{9}-\frac{568218771931153}{28223}a^{8}-\frac{275775996448735}{4342}a^{7}+\frac{48\cdots 23}{1467596}a^{6}-\frac{12\cdots 47}{112892}a^{5}-\frac{34\cdots 79}{56446}a^{4}-\frac{11\cdots 25}{8684}a^{3}+\frac{104610387499017}{4342}a^{2}+\frac{723420679915255}{668}a+\frac{22\cdots 37}{668}$, $\frac{621835766824457}{3224308412}a^{20}-\frac{33425468503955}{124011862}a^{19}-\frac{24003079052703}{248023724}a^{18}+\frac{870170914647}{366899}a^{17}-\frac{50102800520937}{4769687}a^{16}+\frac{12300465298566}{366899}a^{15}-\frac{63353575956765}{733798}a^{14}-\frac{29\cdots 61}{124011862}a^{13}+\frac{32\cdots 07}{9539374}a^{12}+\frac{10\cdots 35}{9539374}a^{11}-\frac{168312344561979}{56446}a^{10}+\frac{96\cdots 11}{733798}a^{9}-\frac{23\cdots 13}{56446}a^{8}+\frac{60\cdots 25}{56446}a^{7}+\frac{18\cdots 13}{1467596}a^{6}-\frac{754236828390939}{28223}a^{5}+\frac{182502807281133}{112892}a^{4}+\frac{60670617586479}{334}a^{3}-\frac{32\cdots 25}{4342}a^{2}+\frac{752911945561953}{334}a-\frac{956233916154837}{167}$, $\frac{10\cdots 89}{3224308412}a^{20}+\frac{222053873638097}{248023724}a^{19}-\frac{431551852638265}{248023724}a^{18}-\frac{6809198076353}{4769687}a^{17}+\frac{32120737609821}{4769687}a^{16}-\frac{1157445513345}{1467596}a^{15}-\frac{31131607974265}{1467596}a^{14}-\frac{85\cdots 35}{124011862}a^{13}-\frac{90\cdots 16}{4769687}a^{12}+\frac{35\cdots 35}{9539374}a^{11}+\frac{11\cdots 24}{366899}a^{10}-\frac{10\cdots 81}{733798}a^{9}+\frac{44079794745348}{28223}a^{8}+\frac{12\cdots 85}{28223}a^{7}+\frac{52\cdots 41}{1467596}a^{6}+\frac{11\cdots 95}{112892}a^{5}-\frac{21\cdots 57}{112892}a^{4}-\frac{34\cdots 98}{2171}a^{3}+\frac{32\cdots 65}{4342}a^{2}-\frac{513996455497371}{668}a-\frac{15\cdots 27}{668}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 50323767605348790 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 50323767605348790 \cdot 1}{2\cdot\sqrt{42600210199743647641790494093640979042918020345647}}\cr\approx \mathstrut & 0.470702043396706 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^21 - 2171*x^14 + 1219335*x^7 - 62748517) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 - 2171*x^14 + 1219335*x^7 - 62748517, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2171*x^14 + 1219335*x^7 - 62748517); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 2171*x^14 + 1219335*x^7 - 62748517); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$F_7$ (as 21T4):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.9544178519053087.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 42
Degree 7 sibling: 7.1.9544178519053087.1
Degree 14 sibling: deg 14
Minimal sibling: 7.1.9544178519053087.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{7}$ ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ R ${\href{/padicField/11.3.0.1}{3} }^{7}$ R ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.3.0.1}{3} }^{7}$ ${\href{/padicField/29.7.0.1}{7} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.7.0.1}{7} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.3.0.1}{3} }^{7}$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.1.21.35a3.1$x^{21} + 21 x^{15} + 7$$21$$1$$35$21T4not computed
\(13\) Copy content Toggle raw display 13.1.7.6a1.1$x^{7} + 13$$7$$1$$6$$D_{7}$$$[\ ]_{7}^{2}$$
13.1.7.6a1.1$x^{7} + 13$$7$$1$$6$$D_{7}$$$[\ ]_{7}^{2}$$
13.1.7.6a1.1$x^{7} + 13$$7$$1$$6$$D_{7}$$$[\ ]_{7}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)