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Properties

Label 12.4.630...936.2
Degree $12$
Signature $[4, 4]$
Discriminant $6.306\times 10^{18}$
Root discriminant \(36.87\)
Ramified primes $2,3,29$
Class number $1$
Class group trivial
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 - 16*x^10 - 36*x^9 + 111*x^8 + 372*x^7 - 204*x^6 - 1500*x^5 - 384*x^4 + 2200*x^3 + 1404*x^2 - 688*x + 21)
 
gp: K = bnfinit(y^12 - 16*y^10 - 36*y^9 + 111*y^8 + 372*y^7 - 204*y^6 - 1500*y^5 - 384*y^4 + 2200*y^3 + 1404*y^2 - 688*y + 21, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 16*x^10 - 36*x^9 + 111*x^8 + 372*x^7 - 204*x^6 - 1500*x^5 - 384*x^4 + 2200*x^3 + 1404*x^2 - 688*x + 21);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 16*x^10 - 36*x^9 + 111*x^8 + 372*x^7 - 204*x^6 - 1500*x^5 - 384*x^4 + 2200*x^3 + 1404*x^2 - 688*x + 21)
 

\( x^{12} - 16 x^{10} - 36 x^{9} + 111 x^{8} + 372 x^{7} - 204 x^{6} - 1500 x^{5} - 384 x^{4} + 2200 x^{3} + \cdots + 21 \) 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:  $[4, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(6306188441142951936\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 29^{4}\) 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:  \(36.87\)
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:  $2^{43/16}3^{25/18}29^{1/2}\approx 159.54585350123654$
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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{45053529185373}a^{11}+\frac{10434425774303}{45053529185373}a^{10}-\frac{4656191672983}{15017843061791}a^{9}+\frac{2035747627828}{15017843061791}a^{8}-\frac{5452823508753}{15017843061791}a^{7}+\frac{6597167961589}{15017843061791}a^{6}+\frac{546979371373}{15017843061791}a^{5}+\frac{866787520750}{15017843061791}a^{4}+\frac{239468823963}{15017843061791}a^{3}-\frac{7834830179465}{45053529185373}a^{2}-\frac{658511320498}{45053529185373}a-\frac{2768899632708}{15017843061791}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

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:  $7$
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:   $\frac{234747129592}{45053529185373}a^{11}+\frac{430028520785}{45053529185373}a^{10}-\frac{895645072869}{15017843061791}a^{9}-\frac{4525865270169}{15017843061791}a^{8}-\frac{764305322438}{15017843061791}a^{7}+\frac{25330859203881}{15017843061791}a^{6}+\frac{38984398876025}{15017843061791}a^{5}-\frac{29564164375469}{15017843061791}a^{4}-\frac{103648773634578}{15017843061791}a^{3}-\frac{168767776252139}{45053529185373}a^{2}+\frac{120616667799071}{45053529185373}a+\frac{9049236738683}{15017843061791}$, $\frac{179579427473}{15017843061791}a^{11}+\frac{66977763602}{15017843061791}a^{10}-\frac{2848369088660}{15017843061791}a^{9}-\frac{7536104621313}{15017843061791}a^{8}+\frac{17244793192210}{15017843061791}a^{7}+\frac{73164599749369}{15017843061791}a^{6}-\frac{10455038120020}{15017843061791}a^{5}-\frac{276313878946645}{15017843061791}a^{4}-\frac{160417526444124}{15017843061791}a^{3}+\frac{352830225685344}{15017843061791}a^{2}+\frac{357522161374313}{15017843061791}a-\frac{25680721598243}{15017843061791}$, $\frac{80438202125}{15017843061791}a^{11}+\frac{264957551563}{15017843061791}a^{10}-\frac{761675759688}{15017843061791}a^{9}-\frac{5999878655411}{15017843061791}a^{8}-\frac{6230082006555}{15017843061791}a^{7}+\frac{27336492810187}{15017843061791}a^{6}+\frac{68471233365819}{15017843061791}a^{5}-\frac{9889346827194}{15017843061791}a^{4}-\frac{141950806710303}{15017843061791}a^{3}-\frac{100459152730511}{15017843061791}a^{2}+\frac{16983112972101}{15017843061791}a-\frac{1227839601109}{15017843061791}$, $\frac{90651048562}{45053529185373}a^{11}-\frac{381354325174}{45053529185373}a^{10}-\frac{701606089890}{15017843061791}a^{9}+\frac{488869983627}{15017843061791}a^{8}+\frac{10674093651220}{15017843061791}a^{7}+\frac{10631228762174}{15017843061791}a^{6}-\frac{51661067582758}{15017843061791}a^{5}-\frac{102368791712110}{15017843061791}a^{4}+\frac{57709399654215}{15017843061791}a^{3}+\frac{710366688677614}{45053529185373}a^{2}+\frac{335620701817565}{45053529185373}a-\frac{57532055003602}{15017843061791}$, $\frac{186233747432}{15017843061791}a^{11}+\frac{418229627661}{15017843061791}a^{10}-\frac{2103659835422}{15017843061791}a^{9}-\frac{11563980372923}{15017843061791}a^{8}-\frac{4429693520318}{15017843061791}a^{7}+\frac{62592574200264}{15017843061791}a^{6}+\frac{102713047199470}{15017843061791}a^{5}-\frac{65091308036849}{15017843061791}a^{4}-\frac{246130957728508}{15017843061791}a^{3}-\frac{139926377357555}{15017843061791}a^{2}+\frac{13534647257900}{15017843061791}a-\frac{16729363505482}{15017843061791}$, $\frac{18096096934}{15017843061791}a^{11}+\frac{277418468303}{15017843061791}a^{10}+\frac{412054241940}{15017843061791}a^{9}-\frac{3961954911344}{15017843061791}a^{8}-\frac{14928330482009}{15017843061791}a^{7}+\frac{61283073176}{15017843061791}a^{6}+\frac{89466512298293}{15017843061791}a^{5}+\frac{90167684799124}{15017843061791}a^{4}-\frac{125171905816727}{15017843061791}a^{3}-\frac{227005578384173}{15017843061791}a^{2}-\frac{34852587822166}{15017843061791}a+\frac{36016907117942}{15017843061791}$, $\frac{1431358504402}{45053529185373}a^{11}+\frac{4091906328542}{45053529185373}a^{10}-\frac{3313021205741}{15017843061791}a^{9}-\frac{20791241373015}{15017843061791}a^{8}-\frac{4102164781328}{15017843061791}a^{7}+\frac{111466912646463}{15017843061791}a^{6}+\frac{77371452665631}{15017843061791}a^{5}-\frac{254940049719087}{15017843061791}a^{4}-\frac{162096550617293}{15017843061791}a^{3}+\frac{814208480694625}{45053529185373}a^{2}-\frac{214792857251281}{45053529185373}a+\frac{2004695177417}{15017843061791}$ 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:  \( 398404.938394 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 398404.938394 \cdot 1}{2\cdot\sqrt{6306188441142951936}}\cr\approx \mathstrut & 1.97811119758 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 16*x^10 - 36*x^9 + 111*x^8 + 372*x^7 - 204*x^6 - 1500*x^5 - 384*x^4 + 2200*x^3 + 1404*x^2 - 688*x + 21)
 
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 - 16*x^10 - 36*x^9 + 111*x^8 + 372*x^7 - 204*x^6 - 1500*x^5 - 384*x^4 + 2200*x^3 + 1404*x^2 - 688*x + 21, 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 - 16*x^10 - 36*x^9 + 111*x^8 + 372*x^7 - 204*x^6 - 1500*x^5 - 384*x^4 + 2200*x^3 + 1404*x^2 - 688*x + 21);
 
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 - 16*x^10 - 36*x^9 + 111*x^8 + 372*x^7 - 204*x^6 - 1500*x^5 - 384*x^4 + 2200*x^3 + 1404*x^2 - 688*x + 21);
 
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: 12.4.6306188441142951936.1

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 ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ R ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.12.24.146$x^{12} + 12 x^{11} + 52 x^{10} + 64 x^{9} + 162 x^{8} + 376 x^{7} + 712 x^{6} + 496 x^{5} + 996 x^{4} + 832 x^{3} + 1520 x^{2} + 800 x + 584$$4$$3$$24$12T60$[2, 2, 2, 3, 3]^{3}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.9.12.21$x^{9} + 6 x^{4} + 3$$9$$1$$12$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
\(29\) Copy content Toggle raw display 29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.2$x^{8} + 1682 x^{4} - 365835 x^{2} + 1414562$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$