↑

Properties

Label 12.12.629...792.1
Degree $12$
Signature $[12, 0]$
Discriminant $6.296\times 10^{23}$
Root discriminant \(96.22\)
Ramified primes $2,3,7,13$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573)
 
gp: K = bnfinit(y^12 - 273*y^10 + 28665*y^8 - 1444716*y^6 + 35395542*y^4 - 371653191*y^2 + 1114959573, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573)
 

\( x^{12} - 273x^{10} + 28665x^{8} - 1444716x^{6} + 35395542x^{4} - 371653191x^{2} + 1114959573 \) 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:  $[12, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(629582418765353043873792\) \(\medspace = 2^{12}\cdot 3^{6}\cdot 7^{6}\cdot 13^{11}\) 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:  \(96.22\)
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:  $2\cdot 3^{1/2}7^{1/2}13^{11/12}\approx 96.21756959272508$
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:  \(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1092=2^{2}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1092}(1,·)$, $\chi_{1092}(839,·)$, $\chi_{1092}(841,·)$, $\chi_{1092}(587,·)$, $\chi_{1092}(589,·)$, $\chi_{1092}(1007,·)$, $\chi_{1092}(337,·)$, $\chi_{1092}(83,·)$, $\chi_{1092}(757,·)$, $\chi_{1092}(673,·)$, $\chi_{1092}(167,·)$, $\chi_{1092}(671,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{21}a^{2}$, $\frac{1}{21}a^{3}$, $\frac{1}{441}a^{4}$, $\frac{1}{441}a^{5}$, $\frac{1}{9261}a^{6}$, $\frac{1}{9261}a^{7}$, $\frac{1}{194481}a^{8}$, $\frac{1}{194481}a^{9}$, $\frac{1}{4084101}a^{10}$, $\frac{1}{4084101}a^{11}$ 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

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
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:   $\frac{1}{9261}a^{6}-\frac{2}{147}a^{4}+\frac{3}{7}a^{2}-2$, $\frac{1}{4084101}a^{10}-\frac{11}{194481}a^{8}+\frac{44}{9261}a^{6}-\frac{11}{63}a^{4}+\frac{55}{21}a^{2}-11$, $\frac{1}{194481}a^{8}-\frac{1}{1029}a^{6}+\frac{3}{49}a^{4}-\frac{29}{21}a^{2}+6$, $\frac{1}{194481}a^{8}-\frac{8}{9261}a^{6}+\frac{20}{441}a^{4}-\frac{16}{21}a^{2}+3$, $\frac{1}{4084101}a^{10}-\frac{11}{194481}a^{8}+\frac{44}{9261}a^{6}-\frac{11}{63}a^{4}+\frac{55}{21}a^{2}-10$, $\frac{1}{4084101}a^{11}-\frac{1}{453789}a^{10}-\frac{13}{194481}a^{9}+\frac{11}{21609}a^{8}+\frac{61}{9261}a^{7}-\frac{407}{9261}a^{6}-\frac{124}{441}a^{5}+\frac{751}{441}a^{4}+\frac{97}{21}a^{3}-\frac{562}{21}a^{2}-11a+82$, $\frac{1}{1361367}a^{11}-\frac{32}{194481}a^{9}+\frac{2}{27783}a^{8}+\frac{2}{147}a^{7}-\frac{2}{147}a^{6}-\frac{74}{147}a^{5}+\frac{52}{63}a^{4}+\frac{23}{3}a^{3}-\frac{52}{3}a^{2}-26a+64$, $\frac{1}{4084101}a^{10}-\frac{4}{194481}a^{9}+\frac{1}{21609}a^{8}+\frac{5}{1323}a^{7}-\frac{125}{9261}a^{6}-\frac{100}{441}a^{5}+\frac{386}{441}a^{4}+\frac{100}{21}a^{3}-\frac{391}{21}a^{2}-19a+69$, $\frac{1}{4084101}a^{11}+\frac{8}{4084101}a^{10}-\frac{10}{194481}a^{9}-\frac{80}{194481}a^{8}+\frac{5}{1323}a^{7}+\frac{40}{1323}a^{6}-\frac{17}{147}a^{5}-\frac{409}{441}a^{4}+\frac{4}{3}a^{3}+\frac{75}{7}a^{2}-5a-35$, $\frac{2}{4084101}a^{11}+\frac{11}{4084101}a^{10}-\frac{22}{194481}a^{9}-\frac{110}{194481}a^{8}+\frac{89}{9261}a^{7}+\frac{376}{9261}a^{6}-\frac{55}{147}a^{5}-\frac{496}{441}a^{4}+\frac{145}{21}a^{3}+\frac{62}{7}a^{2}-51a+35$, $\frac{1}{4084101}a^{11}-\frac{11}{4084101}a^{10}-\frac{10}{194481}a^{9}+\frac{43}{64827}a^{8}+\frac{31}{9261}a^{7}-\frac{548}{9261}a^{6}-\frac{20}{441}a^{5}+\frac{1007}{441}a^{4}-\frac{41}{21}a^{3}-\frac{106}{3}a^{2}+35a+178$ 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:  \( 13423777.933 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 13423777.933 \cdot 4}{2\cdot\sqrt{629582418765353043873792}}\cr\approx \mathstrut & 0.13859207812 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573)
 
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 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573, 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 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573);
 
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 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573);
 
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

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.4.15502032.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/padicField/5.4.0.1}{4} }^{3}$ R ${\href{/padicField/11.12.0.1}{12} }$ R ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.12.0.1}{12} }$

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

# 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 2.12.12.25$x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$$2$$6$$12$$C_{12}$$[2]^{6}$
\(3\) Copy content Toggle raw display 3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(7\) Copy content Toggle raw display 7.12.6.2$x^{12} + 49 x^{8} - 1715 x^{6} + 9604 x^{4} - 100842 x^{2} + 352947$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
\(13\) Copy content Toggle raw display 13.12.11.1$x^{12} + 156$$12$$1$$11$$C_{12}$$[\ ]_{12}$