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Properties

Label 12.0.396...352.1
Degree $12$
Signature $[0, 6]$
Discriminant $3.962\times 10^{23}$
Root discriminant \(92.58\)
Ramified primes $2,13,17$
Class number $1332$ (GRH)
Class group [6, 222] (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 + 41*x^10 - 130*x^9 + 945*x^8 - 2044*x^7 + 12317*x^6 - 19586*x^5 + 105551*x^4 - 77288*x^3 + 634081*x^2 - 13652*x + 1704149)
 
gp: K = bnfinit(y^12 - 4*y^11 + 41*y^10 - 130*y^9 + 945*y^8 - 2044*y^7 + 12317*y^6 - 19586*y^5 + 105551*y^4 - 77288*y^3 + 634081*y^2 - 13652*y + 1704149, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 41*x^10 - 130*x^9 + 945*x^8 - 2044*x^7 + 12317*x^6 - 19586*x^5 + 105551*x^4 - 77288*x^3 + 634081*x^2 - 13652*x + 1704149);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 41*x^10 - 130*x^9 + 945*x^8 - 2044*x^7 + 12317*x^6 - 19586*x^5 + 105551*x^4 - 77288*x^3 + 634081*x^2 - 13652*x + 1704149)
 

\( x^{12} - 4 x^{11} + 41 x^{10} - 130 x^{9} + 945 x^{8} - 2044 x^{7} + 12317 x^{6} - 19586 x^{5} + \cdots + 1704149 \) 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:   \(396229730290715706724352\) \(\medspace = 2^{12}\cdot 13^{8}\cdot 17^{9}\) 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:  \(92.58\)
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:  $2\cdot 13^{2/3}17^{3/4}\approx 92.57539808355045$
Ramified primes:   \(2\), \(13\), \(17\) 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:  \(884=2^{2}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{884}(1,·)$, $\chi_{884}(373,·)$, $\chi_{884}(599,·)$, $\chi_{884}(523,·)$, $\chi_{884}(781,·)$, $\chi_{884}(237,·)$, $\chi_{884}(659,·)$, $\chi_{884}(341,·)$, $\chi_{884}(183,·)$, $\chi_{884}(55,·)$, $\chi_{884}(477,·)$, $\chi_{884}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.78608.1$^{2}$, 12.0.396229730290715706724352.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{37967122}a^{10}-\frac{653596}{18983561}a^{9}+\frac{4289667}{37967122}a^{8}+\frac{16298135}{37967122}a^{7}-\frac{2708460}{18983561}a^{6}+\frac{5896178}{18983561}a^{5}-\frac{8258911}{18983561}a^{4}+\frac{10886001}{37967122}a^{3}+\frac{557468}{18983561}a^{2}+\frac{6580857}{37967122}a+\frac{13999507}{37967122}$, $\frac{1}{19\!\cdots\!18}a^{11}+\frac{2473506753}{19\!\cdots\!18}a^{10}+\frac{13\!\cdots\!73}{19\!\cdots\!18}a^{9}+\frac{31\!\cdots\!87}{95\!\cdots\!09}a^{8}+\frac{68\!\cdots\!09}{19\!\cdots\!18}a^{7}+\frac{27\!\cdots\!19}{95\!\cdots\!09}a^{6}-\frac{24\!\cdots\!24}{95\!\cdots\!09}a^{5}+\frac{47\!\cdots\!69}{19\!\cdots\!18}a^{4}-\frac{10\!\cdots\!75}{19\!\cdots\!18}a^{3}-\frac{89\!\cdots\!61}{19\!\cdots\!18}a^{2}-\frac{42\!\cdots\!51}{95\!\cdots\!09}a+\frac{55\!\cdots\!99}{19\!\cdots\!18}$ 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

$C_{6}\times C_{222}$, which has order $1332$ (assuming GRH)

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

Relative class number: $1332$ (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:   $\frac{76606174412}{95\!\cdots\!09}a^{11}+\frac{260978061466}{95\!\cdots\!09}a^{10}-\frac{2650509508552}{95\!\cdots\!09}a^{9}+\frac{24247638529406}{95\!\cdots\!09}a^{8}-\frac{100392558045800}{95\!\cdots\!09}a^{7}+\frac{576401589760214}{95\!\cdots\!09}a^{6}-\frac{19\!\cdots\!98}{95\!\cdots\!09}a^{5}+\frac{65\!\cdots\!29}{95\!\cdots\!09}a^{4}-\frac{18\!\cdots\!18}{95\!\cdots\!09}a^{3}+\frac{49\!\cdots\!01}{95\!\cdots\!09}a^{2}-\frac{52\!\cdots\!10}{95\!\cdots\!09}a+\frac{14\!\cdots\!97}{95\!\cdots\!09}$, $\frac{245861562124}{95\!\cdots\!09}a^{11}+\frac{1551653461166}{95\!\cdots\!09}a^{10}-\frac{5459097107964}{95\!\cdots\!09}a^{9}+\frac{89921813275141}{95\!\cdots\!09}a^{8}-\frac{240733711485784}{95\!\cdots\!09}a^{7}+\frac{20\!\cdots\!24}{95\!\cdots\!09}a^{6}-\frac{42\!\cdots\!90}{95\!\cdots\!09}a^{5}+\frac{25\!\cdots\!94}{95\!\cdots\!09}a^{4}-\frac{35\!\cdots\!22}{95\!\cdots\!09}a^{3}+\frac{18\!\cdots\!11}{95\!\cdots\!09}a^{2}-\frac{89\!\cdots\!22}{95\!\cdots\!09}a+\frac{83\!\cdots\!43}{95\!\cdots\!09}$, $\frac{1058199343816}{95\!\cdots\!09}a^{11}-\frac{11843703380354}{95\!\cdots\!09}a^{10}+\frac{71063391914484}{95\!\cdots\!09}a^{9}-\frac{369818424259686}{95\!\cdots\!09}a^{8}+\frac{15\!\cdots\!84}{95\!\cdots\!09}a^{7}-\frac{64\!\cdots\!18}{95\!\cdots\!09}a^{6}+\frac{18\!\cdots\!20}{95\!\cdots\!09}a^{5}-\frac{59\!\cdots\!89}{95\!\cdots\!09}a^{4}+\frac{12\!\cdots\!32}{95\!\cdots\!09}a^{3}-\frac{34\!\cdots\!74}{95\!\cdots\!09}a^{2}+\frac{34\!\cdots\!52}{95\!\cdots\!09}a-\frac{14\!\cdots\!72}{95\!\cdots\!09}$, $\frac{981593169404}{95\!\cdots\!09}a^{11}-\frac{12104681441820}{95\!\cdots\!09}a^{10}+\frac{73713901423036}{95\!\cdots\!09}a^{9}-\frac{394066062789092}{95\!\cdots\!09}a^{8}+\frac{16\!\cdots\!84}{95\!\cdots\!09}a^{7}-\frac{69\!\cdots\!32}{95\!\cdots\!09}a^{6}+\frac{20\!\cdots\!18}{95\!\cdots\!09}a^{5}-\frac{66\!\cdots\!18}{95\!\cdots\!09}a^{4}+\frac{14\!\cdots\!50}{95\!\cdots\!09}a^{3}-\frac{39\!\cdots\!75}{95\!\cdots\!09}a^{2}+\frac{40\!\cdots\!62}{95\!\cdots\!09}a-\frac{16\!\cdots\!69}{95\!\cdots\!09}$, $\frac{520}{18983561}a^{11}-\frac{4104}{18983561}a^{10}+\frac{26160}{18983561}a^{9}-\frac{111610}{18983561}a^{8}+\frac{528600}{18983561}a^{7}-\frac{1737052}{18983561}a^{6}+\frac{5492940}{18983561}a^{5}-\frac{13763110}{18983561}a^{4}+\frac{37315380}{18983561}a^{3}-\frac{64520054}{18983561}a^{2}+\frac{103401340}{18983561}a-\frac{191579799}{18983561}$ 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:  \( 3407.79685773 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3407.79685773 \cdot 1332}{2\cdot\sqrt{396229730290715706724352}}\cr\approx \mathstrut & 0.221847024433 \end{aligned}\] (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 + 41*x^10 - 130*x^9 + 945*x^8 - 2044*x^7 + 12317*x^6 - 19586*x^5 + 105551*x^4 - 77288*x^3 + 634081*x^2 - 13652*x + 1704149)
 
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 + 41*x^10 - 130*x^9 + 945*x^8 - 2044*x^7 + 12317*x^6 - 19586*x^5 + 105551*x^4 - 77288*x^3 + 634081*x^2 - 13652*x + 1704149, 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 + 41*x^10 - 130*x^9 + 945*x^8 - 2044*x^7 + 12317*x^6 - 19586*x^5 + 105551*x^4 - 77288*x^3 + 634081*x^2 - 13652*x + 1704149);
 
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 + 41*x^10 - 130*x^9 + 945*x^8 - 2044*x^7 + 12317*x^6 - 19586*x^5 + 105551*x^4 - 77288*x^3 + 634081*x^2 - 13652*x + 1704149);
 
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

\(\Q(\sqrt{17}) \), 3.3.169.1, 4.0.78608.1, 6.6.140320193.1

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 ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.12.0.1}{12} }$ R R ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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}$
\(13\) Copy content Toggle raw display 13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{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}$