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Properties

Label 12.12.400...493.2
Degree $12$
Signature $[12, 0]$
Discriminant $4.003\times 10^{27}$
Root discriminant \(199.62\)
Ramified primes $3,7,13$
Class number $9$ (GRH)
Class group [3, 3] (GRH)
Galois group $C_{12}$ (as 12T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 - 910*x^9 + 22932*x^8 + 133770*x^7 - 551642*x^6 - 5457816*x^5 - 3504774*x^4 + 64316616*x^3 + 180629631*x^2 + 97478199*x - 97103643)
 
Copy content gp:K = bnfinit(y^12 - 273*y^10 - 910*y^9 + 22932*y^8 + 133770*y^7 - 551642*y^6 - 5457816*y^5 - 3504774*y^4 + 64316616*y^3 + 180629631*y^2 + 97478199*y - 97103643, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 273*x^10 - 910*x^9 + 22932*x^8 + 133770*x^7 - 551642*x^6 - 5457816*x^5 - 3504774*x^4 + 64316616*x^3 + 180629631*x^2 + 97478199*x - 97103643);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 273*x^10 - 910*x^9 + 22932*x^8 + 133770*x^7 - 551642*x^6 - 5457816*x^5 - 3504774*x^4 + 64316616*x^3 + 180629631*x^2 + 97478199*x - 97103643)
 

\( x^{12} - 273 x^{10} - 910 x^{9} + 22932 x^{8} + 133770 x^{7} - 551642 x^{6} - 5457816 x^{5} + \cdots - 97103643 \) 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:  $12$
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:  $[12, 0]$
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:   \(4002614648521196621698550493\) \(\medspace = 3^{18}\cdot 7^{8}\cdot 13^{11}\) 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:  \(199.62\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{3/2}7^{2/3}13^{11/12}\approx 199.61598450849033$
Ramified primes:   \(3\), \(7\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{12}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(256,·)$, $\chi_{819}(1,·)$, $\chi_{819}(2,·)$, $\chi_{819}(4,·)$, $\chi_{819}(32,·)$, $\chi_{819}(8,·)$, $\chi_{819}(64,·)$, $\chi_{819}(128,·)$, $\chi_{819}(205,·)$, $\chi_{819}(16,·)$, $\chi_{819}(512,·)$, $\chi_{819}(410,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $\frac{1}{7}a^{3}$, $\frac{1}{7}a^{4}$, $\frac{1}{7}a^{5}$, $\frac{1}{49}a^{6}$, $\frac{1}{49}a^{7}$, $\frac{1}{49}a^{8}$, $\frac{1}{5831}a^{9}-\frac{3}{833}a^{7}+\frac{3}{119}a^{5}-\frac{6}{119}a^{4}+\frac{4}{119}a^{3}+\frac{4}{17}a^{2}-\frac{8}{17}a$, $\frac{1}{29155}a^{10}-\frac{1}{29155}a^{9}+\frac{31}{4165}a^{8}+\frac{37}{4165}a^{7}+\frac{38}{4165}a^{6}+\frac{5}{119}a^{5}-\frac{1}{85}a^{4}-\frac{27}{595}a^{3}+\frac{39}{85}a^{2}-\frac{26}{85}a+\frac{2}{5}$, $\frac{1}{56\cdots 15}a^{11}-\frac{1244157259741}{56\cdots 15}a^{10}+\frac{39457877532977}{56\cdots 15}a^{9}+\frac{240515590117857}{81\cdots 45}a^{8}+\frac{88758580462739}{11\cdots 35}a^{7}-\frac{115721835010506}{16\cdots 69}a^{6}-\frac{5407994235176}{680920056158755}a^{5}+\frac{19449005488659}{680920056158755}a^{4}+\frac{53476767108868}{11\cdots 35}a^{3}-\frac{481594030127251}{16\cdots 05}a^{2}-\frac{431543596847896}{16\cdots 05}a-\frac{22198159315}{318932110613}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{3}\times C_{3}$, which has order $9$ (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:  $C_{6}\times C_{6}$, which has order $36$ (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:   $\frac{355895910}{54460528735501}a^{11}-\frac{2038491864}{54460528735501}a^{10}-\frac{85612736764}{54460528735501}a^{9}+\frac{23941542465}{7780075533643}a^{8}+\frac{1032575417802}{7780075533643}a^{7}+\frac{868566082998}{7780075533643}a^{6}-\frac{279774190215}{65378785997}a^{5}-\frac{725327993871}{65378785997}a^{4}+\frac{46182872472711}{1111439361949}a^{3}+\frac{28907703541197}{158777051707}a^{2}+\frac{20645651693277}{158777051707}a-\frac{16377645655}{153111911}$, $\frac{1986474849759}{56\cdots 15}a^{11}-\frac{10182990801948}{56\cdots 15}a^{10}-\frac{488492710398113}{56\cdots 15}a^{9}+\frac{97459884822429}{81\cdots 45}a^{8}+\frac{59\cdots 24}{81\cdots 45}a^{7}+\frac{76\cdots 98}{81\cdots 45}a^{6}-\frac{16\cdots 24}{680920056158755}a^{5}-\frac{956706554525478}{136184011231751}a^{4}+\frac{52\cdots 28}{23\cdots 67}a^{3}+\frac{36\cdots 09}{330732598705681}a^{2}+\frac{26\cdots 22}{330732598705681}a-\frac{103629609277163}{1594660553065}$, $\frac{2275278297561}{56\cdots 15}a^{11}-\frac{12674936935848}{56\cdots 15}a^{10}-\frac{78847345934103}{81\cdots 45}a^{9}+\frac{29096241855579}{16\cdots 69}a^{8}+\frac{66\cdots 34}{81\cdots 45}a^{7}+\frac{59\cdots 99}{81\cdots 45}a^{6}-\frac{18\cdots 71}{680920056158755}a^{5}-\frac{47\cdots 19}{680920056158755}a^{4}+\frac{29\cdots 07}{11\cdots 35}a^{3}+\frac{18\cdots 01}{16\cdots 05}a^{2}+\frac{12\cdots 56}{16\cdots 05}a-\frac{107201394206239}{1594660553065}$, $\frac{165254804109}{11\cdots 35}a^{11}-\frac{4761802361153}{11\cdots 83}a^{10}-\frac{21\cdots 94}{56\cdots 15}a^{9}-\frac{131200270524622}{81\cdots 45}a^{8}+\frac{54\cdots 54}{16\cdots 69}a^{7}+\frac{72\cdots 33}{81\cdots 45}a^{6}-\frac{73\cdots 01}{680920056158755}a^{5}-\frac{30\cdots 82}{680920056158755}a^{4}+\frac{10\cdots 26}{11\cdots 35}a^{3}+\frac{10\cdots 63}{16\cdots 05}a^{2}+\frac{99\cdots 03}{16\cdots 05}a-\frac{748155654985838}{1594660553065}$, $\frac{1108756660203}{18\cdots 03}a^{11}-\frac{6094780554158}{18\cdots 03}a^{10}-\frac{269041478214085}{18\cdots 03}a^{9}+\frac{66950055313931}{265670448140629}a^{8}+\frac{465679367076162}{37952921162947}a^{7}+\frac{32\cdots 18}{265670448140629}a^{6}-\frac{883355893582476}{2232524774291}a^{5}-\frac{24\cdots 63}{2232524774291}a^{4}+\frac{14\cdots 76}{37952921162947}a^{3}+\frac{94\cdots 70}{5421845880421}a^{2}+\frac{68\cdots 13}{5421845880421}a-\frac{33\cdots 92}{318932110613}$, $\frac{45644742041389}{11\cdots 83}a^{11}-\frac{36038981127222}{16\cdots 69}a^{10}-\frac{11\cdots 16}{11\cdots 83}a^{9}+\frac{27\cdots 52}{16\cdots 69}a^{8}+\frac{13\cdots 98}{16\cdots 69}a^{7}+\frac{13\cdots 62}{16\cdots 69}a^{6}-\frac{36\cdots 50}{136184011231751}a^{5}-\frac{98\cdots 31}{136184011231751}a^{4}+\frac{59\cdots 40}{23\cdots 67}a^{3}+\frac{38\cdots 07}{330732598705681}a^{2}+\frac{27\cdots 86}{330732598705681}a-\frac{22\cdots 22}{318932110613}$, $\frac{1739065697231}{16\cdots 69}a^{11}-\frac{40625790965124}{81\cdots 45}a^{10}-\frac{15\cdots 12}{56\cdots 15}a^{9}+\frac{22\cdots 37}{81\cdots 45}a^{8}+\frac{18\cdots 74}{81\cdots 45}a^{7}+\frac{28\cdots 46}{81\cdots 45}a^{6}-\frac{10\cdots 58}{136184011231751}a^{5}-\frac{22\cdots 56}{97274293736965}a^{4}+\frac{80\cdots 86}{11\cdots 35}a^{3}+\frac{58\cdots 88}{16\cdots 05}a^{2}+\frac{45\cdots 58}{16\cdots 05}a-\frac{35\cdots 96}{1594660553065}$, $\frac{508745186373327}{56\cdots 15}a^{11}-\frac{23\cdots 11}{56\cdots 15}a^{10}-\frac{12\cdots 62}{56\cdots 15}a^{9}+\frac{37\cdots 07}{16\cdots 69}a^{8}+\frac{15\cdots 13}{81\cdots 45}a^{7}+\frac{23\cdots 98}{81\cdots 45}a^{6}-\frac{43\cdots 07}{680920056158755}a^{5}-\frac{13\cdots 58}{680920056158755}a^{4}+\frac{69\cdots 04}{11\cdots 35}a^{3}+\frac{49\cdots 77}{16\cdots 05}a^{2}+\frac{37\cdots 92}{16\cdots 05}a-\frac{29\cdots 73}{1594660553065}$, $\frac{1175303250193}{11\cdots 83}a^{11}-\frac{5684207260406}{81\cdots 45}a^{10}-\frac{13\cdots 23}{56\cdots 15}a^{9}+\frac{544134685297898}{81\cdots 45}a^{8}+\frac{15\cdots 16}{81\cdots 45}a^{7}+\frac{46\cdots 84}{81\cdots 45}a^{6}-\frac{119707277296113}{19454858747393}a^{5}-\frac{13\cdots 94}{97274293736965}a^{4}+\frac{69\cdots 04}{11\cdots 35}a^{3}+\frac{41\cdots 77}{16\cdots 05}a^{2}+\frac{29\cdots 47}{16\cdots 05}a-\frac{241646190912079}{1594660553065}$, $\frac{44112320888357}{56\cdots 15}a^{11}-\frac{56752542872584}{56\cdots 15}a^{10}-\frac{12\cdots 39}{56\cdots 15}a^{9}-\frac{31\cdots 03}{81\cdots 45}a^{8}+\frac{22\cdots 01}{11\cdots 35}a^{7}+\frac{60\cdots 84}{81\cdots 45}a^{6}-\frac{41\cdots 57}{680920056158755}a^{5}-\frac{45\cdots 45}{136184011231751}a^{4}+\frac{10\cdots 11}{23\cdots 67}a^{3}+\frac{14\cdots 67}{330732598705681}a^{2}+\frac{15\cdots 80}{330732598705681}a-\frac{53\cdots 04}{1594660553065}$, $\frac{10470237926839}{11\cdots 83}a^{11}-\frac{26702958028139}{11\cdots 83}a^{10}-\frac{27\cdots 61}{11\cdots 83}a^{9}-\frac{48319605566174}{23\cdots 67}a^{8}+\frac{35\cdots 63}{16\cdots 69}a^{7}+\frac{10\cdots 47}{16\cdots 69}a^{6}-\frac{93\cdots 28}{136184011231751}a^{5}-\frac{44\cdots 44}{136184011231751}a^{4}+\frac{12\cdots 66}{23\cdots 67}a^{3}+\frac{15\cdots 05}{330732598705681}a^{2}+\frac{15\cdots 09}{330732598705681}a-\frac{10\cdots 97}{318932110613}$ 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:  \( 2322316634.755115 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 2322316634.755115 \cdot 9}{2\cdot\sqrt{4002614648521196621698550493}}\cr\approx \mathstrut & 0.676584438194669 \end{aligned}\] (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^12 - 273*x^10 - 910*x^9 + 22932*x^8 + 133770*x^7 - 551642*x^6 - 5457816*x^5 - 3504774*x^4 + 64316616*x^3 + 180629631*x^2 + 97478199*x - 97103643) 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^12 - 273*x^10 - 910*x^9 + 22932*x^8 + 133770*x^7 - 551642*x^6 - 5457816*x^5 - 3504774*x^4 + 64316616*x^3 + 180629631*x^2 + 97478199*x - 97103643, 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^12 - 273*x^10 - 910*x^9 + 22932*x^8 + 133770*x^7 - 551642*x^6 - 5457816*x^5 - 3504774*x^4 + 64316616*x^3 + 180629631*x^2 + 97478199*x - 97103643); 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 = PolynomialRing(QQ); K, a = NumberField(x^12 - 273*x^10 - 910*x^9 + 22932*x^8 + 133770*x^7 - 551642*x^6 - 5457816*x^5 - 3504774*x^4 + 64316616*x^3 + 180629631*x^2 + 97478199*x - 97103643); 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

$C_{12}$ (as 12T1):

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); G, transitive_group_identification(G)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.670761.3, 4.4.19773.1, 6.6.5848964148573.4

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }$ R ${\href{/padicField/5.4.0.1}{4} }^{3}$ R ${\href{/padicField/11.12.0.1}{12} }$ R ${\href{/padicField/17.1.0.1}{1} }^{12}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }$

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
\(3\) Copy content Toggle raw display 3.1.6.9a1.3$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$$[2]_{2}$$
3.1.6.9a1.3$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$$[2]_{2}$$
\(7\) Copy content Toggle raw display 7.4.3.8a1.2$x^{12} + 15 x^{10} + 12 x^{9} + 84 x^{8} + 120 x^{7} + 263 x^{6} + 372 x^{5} + 492 x^{4} + 424 x^{3} + 279 x^{2} + 115 x + 27$$3$$4$$8$$C_{12}$$$[\ ]_{3}^{4}$$
\(13\) Copy content Toggle raw display 13.1.12.11a1.3$x^{12} + 39$$12$$1$$11$$C_{12}$$$[\ ]_{12}$$

Spectrum of ring of integers

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