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Properties

Label 12.12.369...832.1
Degree $12$
Signature $[12, 0]$
Discriminant $3.698\times 10^{17}$
Root discriminant \(29.11\)
Ramified primes $2,3$
Class number $1$
Class group trivial
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 - 24*x^10 - 4*x^9 + 180*x^8 + 36*x^7 - 502*x^6 - 36*x^5 + 501*x^4 - 32*x^3 - 138*x^2 + 36*x - 1)
 
gp: K = bnfinit(y^12 - 24*y^10 - 4*y^9 + 180*y^8 + 36*y^7 - 502*y^6 - 36*y^5 + 501*y^4 - 32*y^3 - 138*y^2 + 36*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 24*x^10 - 4*x^9 + 180*x^8 + 36*x^7 - 502*x^6 - 36*x^5 + 501*x^4 - 32*x^3 - 138*x^2 + 36*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 24*x^10 - 4*x^9 + 180*x^8 + 36*x^7 - 502*x^6 - 36*x^5 + 501*x^4 - 32*x^3 - 138*x^2 + 36*x - 1)
 

\( x^{12} - 24x^{10} - 4x^{9} + 180x^{8} + 36x^{7} - 502x^{6} - 36x^{5} + 501x^{4} - 32x^{3} - 138x^{2} + 36x - 1 \) 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:   \(369768517790072832\) \(\medspace = 2^{33}\cdot 3^{16}\) 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:  \(29.11\)
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:  $2^{11/4}3^{4/3}\approx 29.106779845745038$
Ramified primes:   \(2\), \(3\) 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:  \(144=2^{4}\cdot 3^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{144}(1,·)$, $\chi_{144}(133,·)$, $\chi_{144}(97,·)$, $\chi_{144}(73,·)$, $\chi_{144}(13,·)$, $\chi_{144}(109,·)$, $\chi_{144}(49,·)$, $\chi_{144}(85,·)$, $\chi_{144}(25,·)$, $\chi_{144}(121,·)$, $\chi_{144}(61,·)$, $\chi_{144}(37,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7633}a^{10}+\frac{1693}{7633}a^{9}+\frac{3092}{7633}a^{8}-\frac{2436}{7633}a^{7}-\frac{2553}{7633}a^{6}-\frac{2613}{7633}a^{5}-\frac{779}{7633}a^{4}-\frac{2300}{7633}a^{3}+\frac{2172}{7633}a^{2}+\frac{1134}{7633}a-\frac{1440}{7633}$, $\frac{1}{129761}a^{11}-\frac{5}{129761}a^{10}+\frac{28918}{129761}a^{9}+\frac{21751}{129761}a^{8}+\frac{27221}{129761}a^{7}+\frac{12103}{129761}a^{6}-\frac{52109}{129761}a^{5}-\frac{38232}{129761}a^{4}-\frac{15790}{129761}a^{3}+\frac{7450}{129761}a^{2}-\frac{33988}{129761}a+\frac{2560}{129761}$ 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

Trivial group, which has order $1$

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:   $\frac{23220}{129761}a^{11}+\frac{11978}{129761}a^{10}-\frac{550836}{129761}a^{9}-\frac{374271}{129761}a^{8}+\frac{3975816}{129761}a^{7}+\frac{2838662}{129761}a^{6}-\frac{10085208}{129761}a^{5}-\frac{5877837}{129761}a^{4}+\frac{8343560}{129761}a^{3}+\frac{3758488}{129761}a^{2}-\frac{1384436}{129761}a-\frac{159038}{129761}$, $\frac{28944}{129761}a^{11}+\frac{3996}{129761}a^{10}-\frac{694576}{129761}a^{9}-\frac{212301}{129761}a^{8}+\frac{5186808}{129761}a^{7}+\frac{1779924}{129761}a^{6}-\frac{14265216}{129761}a^{5}-\frac{3202416}{129761}a^{4}+\frac{13750344}{129761}a^{3}+\frac{1433412}{129761}a^{2}-\frac{3319912}{129761}a+\frac{476763}{129761}$, $\frac{108}{449}a^{11}+\frac{42}{449}a^{10}-\frac{2574}{449}a^{9}-\frac{1455}{449}a^{8}+\frac{18864}{449}a^{7}+\frac{11654}{449}a^{6}-\frac{49848}{449}a^{5}-\frac{25533}{449}a^{4}+\frac{45200}{449}a^{3}+\frac{17223}{449}a^{2}-\frac{9600}{449}a+\frac{99}{449}$, $\frac{4962}{129761}a^{11}+\frac{5331}{129761}a^{10}-\frac{121671}{129761}a^{9}-\frac{134877}{129761}a^{8}+\frac{918978}{129761}a^{7}+\frac{882370}{129761}a^{6}-\frac{2539911}{129761}a^{5}-\frac{1417071}{129761}a^{4}+\frac{2718719}{129761}a^{3}+\frac{51723}{129761}a^{2}-\frac{944493}{129761}a+\frac{182798}{129761}$, $\frac{36174}{129761}a^{11}+\frac{17469}{129761}a^{10}-\frac{865557}{129761}a^{9}-\frac{555372}{129761}a^{8}+\frac{6370674}{129761}a^{7}+\frac{4250376}{129761}a^{6}-\frac{16945983}{129761}a^{5}-\frac{8796108}{129761}a^{4}+\frac{15781519}{129761}a^{3}+\frac{5029170}{129761}a^{2}-\frac{3718893}{129761}a+\frac{341170}{129761}$, $\frac{2268}{129761}a^{11}+\frac{8142}{129761}a^{10}-\frac{49310}{129761}a^{9}-\frac{208194}{129761}a^{8}+\frac{264888}{129761}a^{7}+\frac{1588082}{129761}a^{6}-\frac{140856}{129761}a^{5}-\frac{4176621}{129761}a^{4}-\frac{687544}{129761}a^{3}+\frac{3544035}{129761}a^{2}+\frac{545512}{129761}a-\frac{448152}{129761}$, $\frac{16044}{129761}a^{11}+\frac{12107}{129761}a^{10}-\frac{376860}{129761}a^{9}-\frac{342684}{129761}a^{8}+\frac{2650420}{129761}a^{7}+\frac{2458680}{129761}a^{6}-\frac{6369714}{129761}a^{5}-\frac{4720496}{129761}a^{4}+\frac{5086048}{129761}a^{3}+\frac{2277355}{129761}a^{2}-\frac{1233908}{129761}a-\frac{7452}{129761}$, $a$, $\frac{15990}{129761}a^{11}-\frac{1495}{129761}a^{10}-\frac{379855}{129761}a^{9}-\frac{31200}{129761}a^{8}+\frac{2791950}{129761}a^{7}+\frac{368210}{129761}a^{6}-\frac{7404441}{129761}a^{5}-\frac{284145}{129761}a^{4}+\frac{6312385}{129761}a^{3}+\frac{162730}{129761}a^{2}-\frac{985455}{129761}a-\frac{23445}{129761}$, $\frac{7230}{129761}a^{11}+\frac{13473}{129761}a^{10}-\frac{170981}{129761}a^{9}-\frac{343071}{129761}a^{8}+\frac{1183866}{129761}a^{7}+\frac{2470452}{129761}a^{6}-\frac{2680767}{129761}a^{5}-\frac{5593692}{129761}a^{4}+\frac{2031175}{129761}a^{3}+\frac{3595758}{129761}a^{2}-\frac{398981}{129761}a-\frac{135593}{129761}$, $\frac{13329}{129761}a^{11}-\frac{9831}{129761}a^{10}-\frac{298230}{129761}a^{9}+\frac{135260}{129761}a^{8}+\frac{2019451}{129761}a^{7}-\frac{464424}{129761}a^{6}-\frac{4889544}{129761}a^{5}+\frac{616319}{129761}a^{4}+\frac{4156911}{129761}a^{3}-\frac{358101}{129761}a^{2}-\frac{743347}{129761}a+\frac{62128}{129761}$ Copy content Toggle raw display
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:  \( 80910.8856521 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 80910.8856521 \cdot 1}{2\cdot\sqrt{369768517790072832}}\cr\approx \mathstrut & 0.272503409905 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 24*x^10 - 4*x^9 + 180*x^8 + 36*x^7 - 502*x^6 - 36*x^5 + 501*x^4 - 32*x^3 - 138*x^2 + 36*x - 1)
 
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 - 24*x^10 - 4*x^9 + 180*x^8 + 36*x^7 - 502*x^6 - 36*x^5 + 501*x^4 - 32*x^3 - 138*x^2 + 36*x - 1, 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 - 24*x^10 - 4*x^9 + 180*x^8 + 36*x^7 - 502*x^6 - 36*x^5 + 501*x^4 - 32*x^3 - 138*x^2 + 36*x - 1);
 
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 - 24*x^10 - 4*x^9 + 180*x^8 + 36*x^7 - 502*x^6 - 36*x^5 + 501*x^4 - 32*x^3 - 138*x^2 + 36*x - 1);
 
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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{16})^+\), 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 ${\href{/padicField/5.12.0.1}{12} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.1.0.1}{1} }^{12}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.12.0.1}{12} }$

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 3.12.16.14$x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$$3$$4$$16$$C_{12}$$[2]^{4}$