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Properties

Label 12.0.824...192.1
Degree $12$
Signature $[0, 6]$
Discriminant $8.250\times 10^{24}$
Root discriminant \(119.23\)
Ramified primes $2,17,19$
Class number $3924$ (GRH)
Class group [6, 654] (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 + 33*x^10 - 74*x^9 + 741*x^8 - 2188*x^7 + 12781*x^6 - 23894*x^5 + 116843*x^4 - 253700*x^3 + 1179577*x^2 - 1888296*x + 3771709)
 
gp: K = bnfinit(y^12 - 4*y^11 + 33*y^10 - 74*y^9 + 741*y^8 - 2188*y^7 + 12781*y^6 - 23894*y^5 + 116843*y^4 - 253700*y^3 + 1179577*y^2 - 1888296*y + 3771709, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 33*x^10 - 74*x^9 + 741*x^8 - 2188*x^7 + 12781*x^6 - 23894*x^5 + 116843*x^4 - 253700*x^3 + 1179577*x^2 - 1888296*x + 3771709);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 33*x^10 - 74*x^9 + 741*x^8 - 2188*x^7 + 12781*x^6 - 23894*x^5 + 116843*x^4 - 253700*x^3 + 1179577*x^2 - 1888296*x + 3771709)
 

\( x^{12} - 4 x^{11} + 33 x^{10} - 74 x^{9} + 741 x^{8} - 2188 x^{7} + 12781 x^{6} - 23894 x^{5} + \cdots + 3771709 \) 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:   \(8249526994473458677256192\) \(\medspace = 2^{12}\cdot 17^{9}\cdot 19^{8}\) 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:  \(119.23\)
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:  $2\cdot 17^{3/4}19^{2/3}\approx 119.22548213043525$
Ramified primes:   \(2\), \(17\), \(19\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1292=2^{2}\cdot 17\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1292}(1,·)$, $\chi_{1292}(387,·)$, $\chi_{1292}(1189,·)$, $\chi_{1292}(577,·)$, $\chi_{1292}(999,·)$, $\chi_{1292}(273,·)$, $\chi_{1292}(463,·)$, $\chi_{1292}(115,·)$, $\chi_{1292}(305,·)$, $\chi_{1292}(1075,·)$, $\chi_{1292}(885,·)$, $\chi_{1292}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.78608.1$^{2}$, 12.0.8249526994473458677256192.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{102682786}a^{10}+\frac{10831735}{51341393}a^{9}-\frac{14154689}{102682786}a^{8}+\frac{19045501}{102682786}a^{7}+\frac{4521473}{51341393}a^{6}+\frac{1748393}{51341393}a^{5}-\frac{7813859}{51341393}a^{4}-\frac{79085}{1237142}a^{3}-\frac{20864854}{51341393}a^{2}+\frac{1982867}{102682786}a+\frac{48794511}{102682786}$, $\frac{1}{84\!\cdots\!06}a^{11}-\frac{1836440497}{42\!\cdots\!53}a^{10}-\frac{41\!\cdots\!71}{42\!\cdots\!53}a^{9}-\frac{14\!\cdots\!26}{42\!\cdots\!53}a^{8}+\frac{10\!\cdots\!41}{42\!\cdots\!53}a^{7}-\frac{13\!\cdots\!57}{84\!\cdots\!06}a^{6}+\frac{23\!\cdots\!73}{84\!\cdots\!06}a^{5}+\frac{21\!\cdots\!99}{42\!\cdots\!53}a^{4}-\frac{21\!\cdots\!29}{84\!\cdots\!06}a^{3}+\frac{11\!\cdots\!55}{84\!\cdots\!06}a^{2}+\frac{17\!\cdots\!39}{84\!\cdots\!06}a-\frac{41\!\cdots\!91}{84\!\cdots\!06}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{6}\times C_{654}$, which has order $3924$ (assuming GRH)

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

Relative class number: $3924$ (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:   $\frac{24486929828}{42\!\cdots\!53}a^{11}-\frac{2083073814370}{42\!\cdots\!53}a^{10}-\frac{3400845507388}{42\!\cdots\!53}a^{9}-\frac{33351284980948}{42\!\cdots\!53}a^{8}-\frac{92684312599816}{42\!\cdots\!53}a^{7}-\frac{10\!\cdots\!70}{42\!\cdots\!53}a^{6}-\frac{15\!\cdots\!30}{42\!\cdots\!53}a^{5}-\frac{78\!\cdots\!67}{42\!\cdots\!53}a^{4}-\frac{31\!\cdots\!90}{42\!\cdots\!53}a^{3}-\frac{10\!\cdots\!01}{42\!\cdots\!53}a^{2}+\frac{73\!\cdots\!86}{42\!\cdots\!53}a-\frac{76\!\cdots\!04}{42\!\cdots\!53}$, $\frac{418849539456}{42\!\cdots\!53}a^{11}-\frac{11024888809260}{42\!\cdots\!53}a^{10}+\frac{14374290835396}{42\!\cdots\!53}a^{9}-\frac{240625444821807}{42\!\cdots\!53}a^{8}+\frac{216338387020512}{42\!\cdots\!53}a^{7}-\frac{61\!\cdots\!74}{42\!\cdots\!53}a^{6}+\frac{61\!\cdots\!92}{42\!\cdots\!53}a^{5}-\frac{73\!\cdots\!77}{42\!\cdots\!53}a^{4}+\frac{29\!\cdots\!60}{42\!\cdots\!53}a^{3}-\frac{68\!\cdots\!86}{42\!\cdots\!53}a^{2}+\frac{69\!\cdots\!92}{42\!\cdots\!53}a-\frac{48\!\cdots\!64}{42\!\cdots\!53}$, $\frac{4413588659400}{42\!\cdots\!53}a^{11}-\frac{4690134240206}{42\!\cdots\!53}a^{10}+\frac{74051949675360}{42\!\cdots\!53}a^{9}-\frac{52318474598027}{42\!\cdots\!53}a^{8}+\frac{21\!\cdots\!08}{42\!\cdots\!53}a^{7}-\frac{34\!\cdots\!48}{42\!\cdots\!53}a^{6}+\frac{19\!\cdots\!32}{42\!\cdots\!53}a^{5}-\frac{12\!\cdots\!56}{42\!\cdots\!53}a^{4}+\frac{19\!\cdots\!72}{42\!\cdots\!53}a^{3}-\frac{45\!\cdots\!94}{42\!\cdots\!53}a^{2}+\frac{13\!\cdots\!04}{42\!\cdots\!53}a+\frac{89\!\cdots\!31}{42\!\cdots\!53}$, $\frac{3970252190116}{42\!\cdots\!53}a^{11}+\frac{8417828383424}{42\!\cdots\!53}a^{10}+\frac{63078504347352}{42\!\cdots\!53}a^{9}+\frac{221658255204728}{42\!\cdots\!53}a^{8}+\frac{19\!\cdots\!12}{42\!\cdots\!53}a^{7}+\frac{37\!\cdots\!96}{42\!\cdots\!53}a^{6}+\frac{14\!\cdots\!70}{42\!\cdots\!53}a^{5}+\frac{68\!\cdots\!88}{42\!\cdots\!53}a^{4}+\frac{20\!\cdots\!02}{42\!\cdots\!53}a^{3}+\frac{33\!\cdots\!93}{42\!\cdots\!53}a^{2}+\frac{59\!\cdots\!26}{42\!\cdots\!53}a+\frac{52\!\cdots\!40}{42\!\cdots\!53}$, $\frac{1064}{51341393}a^{11}-\frac{632}{51341393}a^{10}+\frac{18776}{51341393}a^{9}-\frac{4598}{51341393}a^{8}+\frac{535688}{51341393}a^{7}-\frac{589636}{51341393}a^{6}+\frac{4999844}{51341393}a^{5}-\frac{1196818}{51341393}a^{4}+\frac{56011844}{51341393}a^{3}-\frac{84044866}{51341393}a^{2}+\frac{312663316}{51341393}a+\frac{454841263}{51341393}$ 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:  \( 5500.24745416 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 5500.24745416 \cdot 3924}{2\cdot\sqrt{8249526994473458677256192}}\cr\approx \mathstrut & 0.231177688973 \end{aligned}\] (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 + 33*x^10 - 74*x^9 + 741*x^8 - 2188*x^7 + 12781*x^6 - 23894*x^5 + 116843*x^4 - 253700*x^3 + 1179577*x^2 - 1888296*x + 3771709)
 
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 + 33*x^10 - 74*x^9 + 741*x^8 - 2188*x^7 + 12781*x^6 - 23894*x^5 + 116843*x^4 - 253700*x^3 + 1179577*x^2 - 1888296*x + 3771709, 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 + 33*x^10 - 74*x^9 + 741*x^8 - 2188*x^7 + 12781*x^6 - 23894*x^5 + 116843*x^4 - 253700*x^3 + 1179577*x^2 - 1888296*x + 3771709);
 
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 + 33*x^10 - 74*x^9 + 741*x^8 - 2188*x^7 + 12781*x^6 - 23894*x^5 + 116843*x^4 - 253700*x^3 + 1179577*x^2 - 1888296*x + 3771709);
 
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

\(\Q(\sqrt{17}) \), 3.3.361.1, 4.0.78608.1, 6.6.640267073.1

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 ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.12.0.1}{12} }$ ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ R R ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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 2.6.6.5$x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$$2$$3$$6$$C_6$$[2]^{3}$
\(17\) Copy content Toggle raw display 17.12.9.1$x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(19\) Copy content Toggle raw display 19.3.2.2$x^{3} + 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} + 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} + 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} + 19$$3$$1$$2$$C_3$$[\ ]_{3}$