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Properties

Label 12.0.646...816.2
Degree $12$
Signature $[0, 6]$
Discriminant $6.470\times 10^{31}$
Root discriminant \(447.62\)
Ramified primes $2,3,7,13$
Class number $2736480$ (GRH)
Class group [2, 2, 2, 342060] (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 354*x^10 - 948*x^9 + 51931*x^8 - 155880*x^7 + 1803608*x^6 + 6044736*x^5 - 246375360*x^4 + 568270912*x^3 + 5010387560*x^2 - 23590792560*x + 30754259836)
 
gp: K = bnfinit(y^12 - 4*y^11 + 354*y^10 - 948*y^9 + 51931*y^8 - 155880*y^7 + 1803608*y^6 + 6044736*y^5 - 246375360*y^4 + 568270912*y^3 + 5010387560*y^2 - 23590792560*y + 30754259836, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 354*x^10 - 948*x^9 + 51931*x^8 - 155880*x^7 + 1803608*x^6 + 6044736*x^5 - 246375360*x^4 + 568270912*x^3 + 5010387560*x^2 - 23590792560*x + 30754259836);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 354*x^10 - 948*x^9 + 51931*x^8 - 155880*x^7 + 1803608*x^6 + 6044736*x^5 - 246375360*x^4 + 568270912*x^3 + 5010387560*x^2 - 23590792560*x + 30754259836)
 

\( x^{12} - 4 x^{11} + 354 x^{10} - 948 x^{9} + 51931 x^{8} - 155880 x^{7} + 1803608 x^{6} + \cdots + 30754259836 \) Copy content Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(64696171405251285666634521378816\) \(\medspace = 2^{33}\cdot 3^{6}\cdot 7^{8}\cdot 13^{11}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(447.62\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{11/4}3^{1/2}7^{2/3}13^{11/12}\approx 447.6169754680814$
Ramified primes:   \(2\), \(3\), \(7\), \(13\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{26}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4368=2^{4}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4368}(1,·)$, $\chi_{4368}(3011,·)$, $\chi_{4368}(289,·)$, $\chi_{4368}(529,·)$, $\chi_{4368}(947,·)$, $\chi_{4368}(2867,·)$, $\chi_{4368}(3481,·)$, $\chi_{4368}(2521,·)$, $\chi_{4368}(3035,·)$, $\chi_{4368}(2459,·)$, $\chi_{4368}(3515,·)$, $\chi_{4368}(1369,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.40495104.3$^{2}$, 12.0.64696171405251285666634521378816.2$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13}a^{4}+\frac{3}{13}a^{3}+\frac{5}{13}a^{2}-\frac{4}{13}a-\frac{4}{13}$, $\frac{1}{13}a^{5}-\frac{4}{13}a^{3}-\frac{6}{13}a^{2}-\frac{5}{13}a-\frac{1}{13}$, $\frac{1}{195}a^{6}-\frac{2}{195}a^{5}-\frac{1}{39}a^{4}-\frac{79}{195}a^{3}-\frac{10}{39}a^{2}-\frac{59}{195}$, $\frac{1}{195}a^{7}+\frac{2}{65}a^{5}+\frac{1}{195}a^{4}+\frac{2}{195}a^{3}+\frac{1}{3}a^{2}+\frac{7}{15}a+\frac{92}{195}$, $\frac{1}{5860920}a^{8}+\frac{229}{1465230}a^{7}+\frac{729}{976820}a^{6}-\frac{13781}{1465230}a^{5}+\frac{33451}{1953640}a^{4}+\frac{60100}{146523}a^{3}+\frac{93664}{244205}a^{2}+\frac{115697}{244205}a-\frac{7277}{34476}$, $\frac{1}{5860920}a^{9}+\frac{3443}{2930460}a^{7}-\frac{61}{86190}a^{6}+\frac{38005}{1172184}a^{5}+\frac{135}{5746}a^{4}+\frac{100713}{244205}a^{3}+\frac{17302}{43095}a^{2}-\frac{114077}{586092}a+\frac{13946}{43095}$, $\frac{1}{679667448720}a^{10}+\frac{10223}{169916862180}a^{9}-\frac{28261}{339833724360}a^{8}+\frac{217778267}{169916862180}a^{7}+\frac{234172153}{679667448720}a^{6}+\frac{582056482}{42479215545}a^{5}-\frac{17991866}{8495843109}a^{4}+\frac{20020422217}{84958431090}a^{3}-\frac{138902301277}{339833724360}a^{2}+\frac{3660053278}{14159738515}a-\frac{44944273}{108642495}$, $\frac{1}{54\!\cdots\!40}a^{11}+\frac{62962667}{90\!\cdots\!40}a^{10}+\frac{407645863445}{27\!\cdots\!32}a^{9}+\frac{850345423713}{18\!\cdots\!88}a^{8}-\frac{24\!\cdots\!27}{10\!\cdots\!28}a^{7}-\frac{39\!\cdots\!67}{27\!\cdots\!20}a^{6}+\frac{11\!\cdots\!17}{69\!\cdots\!80}a^{5}+\frac{15\!\cdots\!95}{54\!\cdots\!64}a^{4}-\frac{13\!\cdots\!41}{27\!\cdots\!20}a^{3}-\frac{94\!\cdots\!45}{90\!\cdots\!44}a^{2}-\frac{11\!\cdots\!41}{26\!\cdots\!60}a-\frac{579056113237333}{40\!\cdots\!56}$ Copy content Toggle raw display

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

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{342060}$, which has order $2736480$ (assuming GRH)

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

Relative class number: $456080$ (assuming GRH)

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{56}{12276493292505}a^{11}+\frac{465}{3273731544668}a^{10}+\frac{23039}{24552986585010}a^{9}+\frac{864473}{24552986585010}a^{8}+\frac{1295584}{12276493292505}a^{7}+\frac{5253889}{2135042311740}a^{6}-\frac{120644949}{8184328861670}a^{5}-\frac{2467526531}{12276493292505}a^{4}+\frac{11366684582}{12276493292505}a^{3}+\frac{15502292935}{4910597317002}a^{2}-\frac{1140939133}{55549743405}a+\frac{6235452201}{615640805}$, $\frac{3566751046493}{27\!\cdots\!20}a^{11}+\frac{63730016963}{90\!\cdots\!40}a^{10}+\frac{629452060251377}{13\!\cdots\!60}a^{9}+\frac{212616525387442}{33\!\cdots\!15}a^{8}+\frac{14\!\cdots\!33}{20\!\cdots\!40}a^{7}+\frac{51\!\cdots\!91}{69\!\cdots\!80}a^{6}+\frac{17\!\cdots\!70}{67\!\cdots\!83}a^{5}+\frac{18\!\cdots\!31}{10\!\cdots\!20}a^{4}-\frac{35\!\cdots\!37}{13\!\cdots\!60}a^{3}-\frac{52\!\cdots\!17}{13\!\cdots\!60}a^{2}+\frac{10\!\cdots\!16}{19\!\cdots\!95}a-\frac{31\!\cdots\!77}{33\!\cdots\!30}$, $\frac{29615999215}{54\!\cdots\!64}a^{11}-\frac{461026056989}{90\!\cdots\!40}a^{10}+\frac{7568719302848}{33\!\cdots\!15}a^{9}-\frac{14097092404397}{90\!\cdots\!44}a^{8}+\frac{656120478973109}{18\!\cdots\!88}a^{7}-\frac{66\!\cdots\!19}{27\!\cdots\!20}a^{6}+\frac{54\!\cdots\!77}{27\!\cdots\!32}a^{5}-\frac{72\!\cdots\!60}{22\!\cdots\!61}a^{4}-\frac{44\!\cdots\!87}{27\!\cdots\!32}a^{3}+\frac{11\!\cdots\!65}{90\!\cdots\!44}a^{2}-\frac{14\!\cdots\!41}{39\!\cdots\!90}a+\frac{13\!\cdots\!83}{339573062043163}$, $\frac{4548212423}{13\!\cdots\!76}a^{11}+\frac{11071937221}{18\!\cdots\!88}a^{10}+\frac{10437306400177}{90\!\cdots\!44}a^{9}+\frac{7708715258683}{45\!\cdots\!22}a^{8}+\frac{31\!\cdots\!11}{18\!\cdots\!88}a^{7}+\frac{36\!\cdots\!05}{18\!\cdots\!88}a^{6}+\frac{28\!\cdots\!11}{45\!\cdots\!22}a^{5}+\frac{39\!\cdots\!53}{90\!\cdots\!44}a^{4}-\frac{58\!\cdots\!63}{90\!\cdots\!44}a^{3}-\frac{93\!\cdots\!21}{90\!\cdots\!44}a^{2}+\frac{17\!\cdots\!21}{13\!\cdots\!33}a-\frac{15\!\cdots\!01}{679146124086326}$, $\frac{44865616533}{18\!\cdots\!88}a^{11}+\frac{29145941451}{18\!\cdots\!88}a^{10}+\frac{7874756035487}{90\!\cdots\!44}a^{9}+\frac{7728327322617}{45\!\cdots\!22}a^{8}+\frac{23\!\cdots\!93}{18\!\cdots\!88}a^{7}+\frac{38\!\cdots\!75}{18\!\cdots\!88}a^{6}+\frac{20\!\cdots\!01}{45\!\cdots\!22}a^{5}+\frac{24\!\cdots\!75}{69\!\cdots\!88}a^{4}-\frac{43\!\cdots\!77}{90\!\cdots\!44}a^{3}-\frac{99\!\cdots\!07}{90\!\cdots\!44}a^{2}+\frac{14\!\cdots\!19}{13\!\cdots\!33}a+\frac{74461075246161}{52242009545102}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 280406.634104081 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 280406.634104081 \cdot 2736480}{2\cdot\sqrt{64696171405251285666634521378816}}\cr\approx \mathstrut & 2.93488093339464 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 354*x^10 - 948*x^9 + 51931*x^8 - 155880*x^7 + 1803608*x^6 + 6044736*x^5 - 246375360*x^4 + 568270912*x^3 + 5010387560*x^2 - 23590792560*x + 30754259836)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^12 - 4*x^11 + 354*x^10 - 948*x^9 + 51931*x^8 - 155880*x^7 + 1803608*x^6 + 6044736*x^5 - 246375360*x^4 + 568270912*x^3 + 5010387560*x^2 - 23590792560*x + 30754259836, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^12 - 4*x^11 + 354*x^10 - 948*x^9 + 51931*x^8 - 155880*x^7 + 1803608*x^6 + 6044736*x^5 - 246375360*x^4 + 568270912*x^3 + 5010387560*x^2 - 23590792560*x + 30754259836);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 354*x^10 - 948*x^9 + 51931*x^8 - 155880*x^7 + 1803608*x^6 + 6044736*x^5 - 246375360*x^4 + 568270912*x^3 + 5010387560*x^2 - 23590792560*x + 30754259836);
 
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):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
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

\(\Q(\sqrt{26}) \), 3.3.8281.1, 4.0.40495104.3, 6.6.456434940416.2

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
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 R R ${\href{/padicField/5.6.0.1}{6} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}$ R ${\href{/padicField/17.1.0.1}{1} }^{12}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.1.0.1}{1} }^{12}$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
\(2\) Copy content Toggle raw display 2.4.11.9$x^{4} + 8 x^{3} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.9$x^{4} + 8 x^{3} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.9$x^{4} + 8 x^{3} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
\(3\) Copy content Toggle raw display 3.12.6.1$x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
\(7\) Copy content Toggle raw display 7.12.8.2$x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
\(13\) Copy content Toggle raw display 13.12.11.11$x^{12} + 65$$12$$1$$11$$C_{12}$$[\ ]_{12}$