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Properties

Label 12.0.944...296.1
Degree $12$
Signature $[0, 6]$
Discriminant $9.441\times 10^{18}$
Root discriminant \(38.13\)
Ramified primes $2,3,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $M_{12}$ (as 12T295)

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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 + 4*x^10 + 12*x^9 - 72*x^8 + 168*x^7 - 132*x^6 - 324*x^5 + 1197*x^4 - 1752*x^3 + 1500*x^2 - 672*x + 207)
 
gp: K = bnfinit(y^12 - 4*y^11 + 4*y^10 + 12*y^9 - 72*y^8 + 168*y^7 - 132*y^6 - 324*y^5 + 1197*y^4 - 1752*y^3 + 1500*y^2 - 672*y + 207, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 4*x^10 + 12*x^9 - 72*x^8 + 168*x^7 - 132*x^6 - 324*x^5 + 1197*x^4 - 1752*x^3 + 1500*x^2 - 672*x + 207);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 4*x^10 + 12*x^9 - 72*x^8 + 168*x^7 - 132*x^6 - 324*x^5 + 1197*x^4 - 1752*x^3 + 1500*x^2 - 672*x + 207)
 

\( x^{12} - 4 x^{11} + 4 x^{10} + 12 x^{9} - 72 x^{8} + 168 x^{7} - 132 x^{6} - 324 x^{5} + 1197 x^{4} + \cdots + 207 \) 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:   \(9440732714731831296\) \(\medspace = 2^{24}\cdot 3^{14}\cdot 7^{6}\) 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:  \(38.13\)
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:  $2^{21/8}3^{37/24}7^{2/3}\approx 122.79038648593725$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{20238231435}a^{11}-\frac{12999637}{20238231435}a^{10}-\frac{2769995312}{20238231435}a^{9}-\frac{498670084}{1349215429}a^{8}-\frac{3043916447}{6746077145}a^{7}-\frac{2852555996}{6746077145}a^{6}-\frac{237967280}{1349215429}a^{5}-\frac{35993955}{1349215429}a^{4}-\frac{2575439313}{6746077145}a^{3}-\frac{377860847}{6746077145}a^{2}-\frac{239337899}{1349215429}a-\frac{1154832626}{6746077145}$ 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 $1$ (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:   $\frac{1382528}{1349215429}a^{11}-\frac{15493993}{6746077145}a^{10}-\frac{25747707}{6746077145}a^{9}+\frac{159347749}{6746077145}a^{8}-\frac{163003636}{6746077145}a^{7}-\frac{97579978}{6746077145}a^{6}+\frac{1675288661}{6746077145}a^{5}-\frac{425883629}{6746077145}a^{4}-\frac{3105696354}{6746077145}a^{3}+\frac{9003684558}{6746077145}a^{2}-\frac{4399437169}{6746077145}a+\frac{2255088161}{6746077145}$, $\frac{11968309}{6746077145}a^{11}-\frac{244524127}{6746077145}a^{10}+\frac{741318636}{6746077145}a^{9}-\frac{366082063}{6746077145}a^{8}-\frac{3516203592}{6746077145}a^{7}+\frac{15967982879}{6746077145}a^{6}-\frac{29442521847}{6746077145}a^{5}+\frac{5265036278}{6746077145}a^{4}+\frac{98600220097}{6746077145}a^{3}-\frac{46282896518}{1349215429}a^{2}+\frac{243688728763}{6746077145}a-\frac{123601675169}{6746077145}$, $\frac{105967298}{6746077145}a^{11}-\frac{947016116}{6746077145}a^{10}+\frac{1512876709}{6746077145}a^{9}+\frac{675607999}{1349215429}a^{8}-\frac{13015784748}{6746077145}a^{7}+\frac{40496814471}{6746077145}a^{6}-\frac{7062653262}{1349215429}a^{5}-\frac{17716414966}{1349215429}a^{4}+\frac{279914308928}{6746077145}a^{3}-\frac{363004819403}{6746077145}a^{2}+\frac{36112906485}{1349215429}a-\frac{71925893044}{6746077145}$, $\frac{268268778}{6746077145}a^{11}-\frac{2707817833}{6746077145}a^{10}+\frac{6608871956}{6746077145}a^{9}+\frac{699676701}{6746077145}a^{8}-\frac{42405298747}{6746077145}a^{7}+\frac{149291367199}{6746077145}a^{6}-\frac{237833846466}{6746077145}a^{5}-\frac{34401568626}{6746077145}a^{4}+\frac{956015719307}{6746077145}a^{3}-\frac{2057235070491}{6746077145}a^{2}+\frac{2072336779624}{6746077145}a-\frac{207091455275}{1349215429}$, $\frac{877438313}{6746077145}a^{11}-\frac{3035676632}{6746077145}a^{10}+\frac{59823523}{1349215429}a^{9}+\frac{13986217313}{6746077145}a^{8}-\frac{10236998366}{1349215429}a^{7}+\frac{19471570888}{1349215429}a^{6}-\frac{1089430328}{6746077145}a^{5}-\frac{345471910928}{6746077145}a^{4}+\frac{146140355674}{1349215429}a^{3}-\frac{723992343867}{6746077145}a^{2}+\frac{348109056592}{6746077145}a-\frac{115932287257}{6746077145}$ 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:  \( 130780.174411 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 130780.174411 \cdot 1}{2\cdot\sqrt{9440732714731831296}}\cr\approx \mathstrut & 1.30944805723 \end{aligned}\] (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 + 4*x^10 + 12*x^9 - 72*x^8 + 168*x^7 - 132*x^6 - 324*x^5 + 1197*x^4 - 1752*x^3 + 1500*x^2 - 672*x + 207)
 
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 + 4*x^10 + 12*x^9 - 72*x^8 + 168*x^7 - 132*x^6 - 324*x^5 + 1197*x^4 - 1752*x^3 + 1500*x^2 - 672*x + 207, 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 + 4*x^10 + 12*x^9 - 72*x^8 + 168*x^7 - 132*x^6 - 324*x^5 + 1197*x^4 - 1752*x^3 + 1500*x^2 - 672*x + 207);
 
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 + 4*x^10 + 12*x^9 - 72*x^8 + 168*x^7 - 132*x^6 - 324*x^5 + 1197*x^4 - 1752*x^3 + 1500*x^2 - 672*x + 207);
 
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

$M_{12}$ (as 12T295):

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 non-solvable group of order 95040
The 15 conjugacy class representatives for $M_{12}$
Character table for $M_{12}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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]
 

Sibling fields

Degree 12 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ R ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.12.24.283$x^{12} + 10 x^{10} + 28 x^{9} + 74 x^{8} + 80 x^{7} + 616 x^{6} + 1760 x^{5} + 3820 x^{4} + 4672 x^{3} + 4760 x^{2} + 3184 x + 1784$$4$$3$$24$12T60$[2, 2, 3, 3]^{6}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.13.6$x^{9} + 3 x^{5} + 3$$9$$1$$13$$(C_3^2:C_8):C_2$$[13/8, 13/8]_{8}^{2}$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$