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Properties

Label 12.0.197...648.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.980\times 10^{27}$
Root discriminant \(188.24\)
Ramified primes $2,3,29$
Class number $1066468$ (GRH)
Class group [2, 533234] (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 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112)
 
gp: K = bnfinit(y^12 + 696*y^10 + 151380*y^8 + 13462728*y^6 + 534704436*y^4 + 8860816368*y^2 + 42827279112, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112)
 

\( x^{12} + 696x^{10} + 151380x^{8} + 13462728x^{6} + 534704436x^{4} + 8860816368x^{2} + 42827279112 \) 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:   \(1979522439778248324859035648\) \(\medspace = 2^{33}\cdot 3^{18}\cdot 29^{6}\) 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:  \(188.24\)
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:  $2^{11/4}3^{3/2}29^{1/2}\approx 188.2406306269193$
Ramified primes:   \(2\), \(3\), \(29\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4176=2^{4}\cdot 3^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4176}(1,·)$, $\chi_{4176}(3653,·)$, $\chi_{4176}(2785,·)$, $\chi_{4176}(2089,·)$, $\chi_{4176}(2957,·)$, $\chi_{4176}(173,·)$, $\chi_{4176}(1393,·)$, $\chi_{4176}(2261,·)$, $\chi_{4176}(3481,·)$, $\chi_{4176}(697,·)$, $\chi_{4176}(1565,·)$, $\chi_{4176}(869,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.15501312.6$^{2}$, 12.0.1979522439778248324859035648.1$^{30}$

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

$1$, $a$, $\frac{1}{29}a^{2}$, $\frac{1}{29}a^{3}$, $\frac{1}{1682}a^{4}$, $\frac{1}{1682}a^{5}$, $\frac{1}{146334}a^{6}$, $\frac{1}{146334}a^{7}$, $\frac{1}{8487372}a^{8}$, $\frac{1}{8487372}a^{9}$, $\frac{1}{4184274396}a^{10}-\frac{7}{144285324}a^{8}+\frac{1}{829226}a^{6}-\frac{7}{28594}a^{4}-\frac{7}{493}a^{2}-\frac{2}{17}$, $\frac{1}{4184274396}a^{11}-\frac{7}{144285324}a^{9}+\frac{1}{829226}a^{7}-\frac{7}{28594}a^{5}-\frac{7}{493}a^{3}-\frac{2}{17}a$ 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

$C_{2}\times C_{533234}$, which has order $1066468$ (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:  $5$
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:   $\frac{5}{2092137198}a^{10}+\frac{59}{36071331}a^{8}+\frac{141}{414613}a^{6}+\frac{763}{28594}a^{4}+\frac{372}{493}a^{2}+\frac{82}{17}$, $\frac{3}{697379066}a^{10}+\frac{401}{144285324}a^{8}+\frac{213}{414613}a^{6}+\frac{945}{28594}a^{4}+\frac{384}{493}a^{2}+\frac{100}{17}$, $\frac{11}{2092137198}a^{10}+\frac{41}{12023777}a^{8}+\frac{1579}{2487678}a^{6}+\frac{586}{14297}a^{4}+\frac{441}{493}a^{2}+\frac{41}{17}$, $\frac{1}{697379066}a^{10}+\frac{5}{4975356}a^{8}+\frac{545}{2487678}a^{6}+\frac{268}{14297}a^{4}+\frac{315}{493}a^{2}+\frac{124}{17}$, $\frac{10}{1046068599}a^{10}+\frac{893}{144285324}a^{8}+\frac{2857}{2487678}a^{6}+\frac{73}{986}a^{4}+\frac{825}{493}a^{2}+\frac{158}{17}$ 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:  \( 481.70037561485367 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 481.70037561485367 \cdot 1066468}{2\cdot\sqrt{1979522439778248324859035648}}\cr\approx \mathstrut & 0.355217058431051 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112)
 
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 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112, 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 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112);
 
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 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112);
 
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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 4.0.15501312.6, 6.6.3359232.1

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 R ${\href{/padicField/5.12.0.1}{12} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.1.0.1}{1} }^{12}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ R ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.12.0.1}{12} }$

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 2.12.33.375$x^{12} + 80 x^{10} - 264 x^{9} - 638 x^{8} + 64 x^{7} + 208 x^{6} + 3904 x^{5} + 8348 x^{4} + 10496 x^{3} + 13152 x^{2} + 7200 x + 6392$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
\(3\) Copy content Toggle raw display 3.12.18.74$x^{12} + 12 x^{11} + 42 x^{10} + 42 x^{9} + 54 x^{8} + 18 x^{7} - 33 x^{6} - 252 x^{5} - 126 x^{3} + 882$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
\(29\) Copy content Toggle raw display 29.12.6.2$x^{12} + 841 x^{8} - 609725 x^{6} + 12023777 x^{4} - 266644937 x^{2} + 1189646642$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$