Properties

Label 12.12.111...856.1
Degree $12$
Signature $[12, 0]$
Discriminant $1.115\times 10^{41}$
Root discriminant \(2634.01\)
Ramified primes $2,3,11,197$
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 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608)
 
gp: K = bnfinit(y^12 - 968*y^10 + 346060*y^8 - 56221440*y^6 + 4128059232*y^4 - 247374336*y^3 - 114286943232*y^2 + 19295198208*y + 632319724608, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608)
 

\( x^{12} - 968 x^{10} + 346060 x^{8} - 56221440 x^{6} + 4128059232 x^{4} - 247374336 x^{3} - 114286943232 x^{2} + 19295198208 x + 632319724608 \) 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:   \(111534324540107369492059712937081339641856\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 11^{20}\cdot 197^{4}\) 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:  \(2634.01\)
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:  $2^{43/16}3^{7/8}11^{20/11}197^{1/2}\approx 18500.07845030721$
Ramified primes:   \(2\), \(3\), \(11\), \(197\) 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 \(a\))

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{22}a^{3}$, $\frac{1}{44}a^{4}$, $\frac{1}{484}a^{5}$, $\frac{1}{968}a^{6}$, $\frac{1}{21296}a^{7}-\frac{1}{968}a^{5}-\frac{1}{2}a$, $\frac{1}{276848}a^{8}-\frac{5}{276848}a^{7}-\frac{3}{12584}a^{6}+\frac{1}{1144}a^{5}-\frac{1}{572}a^{4}+\frac{5}{286}a^{3}+\frac{3}{26}a^{2}+\frac{1}{26}a$, $\frac{1}{6090656}a^{9}+\frac{1}{12584}a^{6}-\frac{5}{12584}a^{5}+\frac{1}{286}a^{4}-\frac{3}{572}a^{3}-\frac{2}{13}a^{2}-\frac{11}{26}a$, $\frac{1}{103541152}a^{10}+\frac{1}{12942644}a^{9}-\frac{1}{2353208}a^{8}-\frac{5}{276848}a^{7}+\frac{6}{26741}a^{6}-\frac{109}{213928}a^{5}-\frac{3}{286}a^{4}-\frac{46}{2431}a^{3}+\frac{21}{221}a^{2}-\frac{3}{26}a-\frac{5}{17}$, $\frac{1}{61\!\cdots\!04}a^{11}-\frac{6112003}{28\!\cdots\!32}a^{10}+\frac{72331165}{14\!\cdots\!16}a^{9}-\frac{33810641}{127560303698256}a^{8}-\frac{12724963}{3270777017904}a^{7}+\frac{178788053}{966365937108}a^{6}-\frac{153608237}{644243958072}a^{5}+\frac{6829369}{2662165116}a^{4}+\frac{90450011}{9761272092}a^{3}-\frac{499045985}{1996623837}a^{2}-\frac{114366082}{1996623837}a+\frac{22247857}{153586449}$ 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 $1$ (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:   $\frac{6774911265461}{25\!\cdots\!96}a^{11}-\frac{43786837078279}{935442227120544}a^{10}-\frac{803745684348251}{467721113560272}a^{9}+\frac{12\!\cdots\!43}{42520101232752}a^{8}+\frac{26\!\cdots\!27}{7086683538792}a^{7}-\frac{192307474288877}{29283816276}a^{6}-\frac{846553275838714}{26843498253}a^{5}+\frac{13\!\cdots\!42}{2440318023}a^{4}+\frac{737641189453826}{813439341}a^{3}-\frac{22\!\cdots\!29}{1331082558}a^{2}-\frac{122396976163219}{51195483}a+\frac{48\!\cdots\!44}{51195483}$, $\frac{17182472009}{36\!\cdots\!12}a^{11}-\frac{3183853}{288597972168}a^{10}-\frac{486879121799}{165078040080096}a^{9}+\frac{4673070067}{937943409546}a^{8}+\frac{1352446416697}{2501182425456}a^{7}-\frac{108012053329}{113690110248}a^{6}-\frac{11138312707}{430644357}a^{5}+\frac{53202400051}{430644357}a^{4}-\frac{11275781756}{11042163}a^{3}-\frac{1001275578715}{117448461}a^{2}+\frac{1098424199501}{18068994}a+\frac{1688229794057}{9034497}$, $\frac{39891587461373}{61\!\cdots\!04}a^{11}-\frac{19083084691499}{28\!\cdots\!32}a^{10}-\frac{15\!\cdots\!83}{28\!\cdots\!32}a^{9}+\frac{67053840375835}{11596391245296}a^{8}+\frac{68\!\cdots\!85}{42520101232752}a^{7}-\frac{32\!\cdots\!21}{1932731874216}a^{6}-\frac{14\!\cdots\!09}{80530494759}a^{5}+\frac{54\!\cdots\!45}{29283816276}a^{4}+\frac{31\!\cdots\!93}{4880636046}a^{3}-\frac{13\!\cdots\!80}{1996623837}a^{2}-\frac{16\!\cdots\!09}{3993247674}a+\frac{63\!\cdots\!61}{153586449}$, $\frac{75394044722291}{68\!\cdots\!56}a^{11}-\frac{5458496956885}{28346734155168}a^{10}-\frac{171977679413381}{23985698131296}a^{9}+\frac{17\!\cdots\!15}{14173367077584}a^{8}+\frac{73\!\cdots\!01}{4724455692528}a^{7}-\frac{28\!\cdots\!77}{107373993012}a^{6}-\frac{49\!\cdots\!47}{35791331004}a^{5}+\frac{36\!\cdots\!81}{1626878682}a^{4}+\frac{24\!\cdots\!93}{542292894}a^{3}-\frac{29\!\cdots\!69}{443694186}a^{2}-\frac{12\!\cdots\!41}{443694186}a+\frac{56\!\cdots\!91}{17065161}$, $\frac{112687924600997}{15\!\cdots\!76}a^{11}-\frac{92004429225209}{701581670340408}a^{10}-\frac{13\!\cdots\!93}{28\!\cdots\!32}a^{9}+\frac{980599381496737}{11596391245296}a^{8}+\frac{26\!\cdots\!35}{2657506327047}a^{7}-\frac{34\!\cdots\!93}{1932731874216}a^{6}-\frac{55\!\cdots\!09}{644243958072}a^{5}+\frac{11\!\cdots\!61}{7320954069}a^{4}+\frac{24\!\cdots\!79}{9761272092}a^{3}-\frac{14\!\cdots\!59}{307172898}a^{2}-\frac{26\!\cdots\!83}{3993247674}a+\frac{39\!\cdots\!19}{153586449}$, $\frac{18\!\cdots\!75}{15\!\cdots\!76}a^{11}-\frac{17\!\cdots\!11}{14\!\cdots\!16}a^{10}-\frac{12\!\cdots\!95}{127560303698256}a^{9}+\frac{13\!\cdots\!59}{127560303698256}a^{8}+\frac{94\!\cdots\!07}{3270777017904}a^{7}-\frac{30\!\cdots\!61}{966365937108}a^{6}-\frac{20\!\cdots\!55}{644243958072}a^{5}+\frac{25\!\cdots\!01}{7320954069}a^{4}+\frac{26\!\cdots\!51}{2440318023}a^{3}-\frac{23\!\cdots\!41}{1996623837}a^{2}-\frac{16\!\cdots\!03}{3993247674}a+\frac{10\!\cdots\!05}{153586449}$, $\frac{285292839131}{30\!\cdots\!52}a^{11}+\frac{266151168175}{14\!\cdots\!16}a^{10}-\frac{9401127398383}{28\!\cdots\!32}a^{9}-\frac{1475970593639}{9812331053712}a^{8}-\frac{10546622743351}{10630025308188}a^{7}+\frac{70087782859867}{1932731874216}a^{6}+\frac{328976359869589}{644243958072}a^{5}-\frac{70085640844513}{29283816276}a^{4}-\frac{533259898978855}{9761272092}a^{3}-\frac{60225385311290}{1996623837}a^{2}+\frac{60\!\cdots\!79}{3993247674}a+\frac{508483517324281}{153586449}$, $\frac{148068286815589}{68\!\cdots\!56}a^{11}+\frac{353875741435}{2180518011936}a^{10}-\frac{60\!\cdots\!39}{311814075706848}a^{9}-\frac{20\!\cdots\!07}{14173367077584}a^{8}+\frac{29\!\cdots\!93}{4724455692528}a^{7}+\frac{98\!\cdots\!29}{214747986024}a^{6}-\frac{15\!\cdots\!27}{17895665502}a^{5}-\frac{17\!\cdots\!35}{295796124}a^{4}+\frac{11\!\cdots\!36}{271146447}a^{3}+\frac{66\!\cdots\!47}{221847093}a^{2}-\frac{11\!\cdots\!61}{34130322}a-\frac{32\!\cdots\!74}{17065161}$, $\frac{15\!\cdots\!39}{61\!\cdots\!04}a^{11}+\frac{11\!\cdots\!87}{701581670340408}a^{10}-\frac{63\!\cdots\!25}{28\!\cdots\!32}a^{9}-\frac{94\!\cdots\!99}{63780151849128}a^{8}+\frac{15\!\cdots\!37}{21260050616376}a^{7}+\frac{11\!\cdots\!68}{241591484277}a^{6}-\frac{67\!\cdots\!67}{644243958072}a^{5}-\frac{20\!\cdots\!73}{29283816276}a^{4}+\frac{26\!\cdots\!11}{4880636046}a^{3}+\frac{14\!\cdots\!75}{3993247674}a^{2}-\frac{84\!\cdots\!15}{1996623837}a-\frac{35\!\cdots\!09}{153586449}$, $\frac{45\!\cdots\!93}{30\!\cdots\!52}a^{11}+\frac{76\!\cdots\!19}{14\!\cdots\!16}a^{10}-\frac{39\!\cdots\!63}{28\!\cdots\!32}a^{9}-\frac{65\!\cdots\!29}{127560303698256}a^{8}+\frac{20\!\cdots\!73}{42520101232752}a^{7}+\frac{17\!\cdots\!91}{966365937108}a^{6}-\frac{10\!\cdots\!67}{14641908138}a^{5}-\frac{38\!\cdots\!93}{14641908138}a^{4}+\frac{36\!\cdots\!55}{750867084}a^{3}+\frac{32\!\cdots\!78}{1996623837}a^{2}-\frac{20\!\cdots\!64}{1996623837}a-\frac{43\!\cdots\!77}{153586449}$, $\frac{14\!\cdots\!39}{30\!\cdots\!52}a^{11}-\frac{42\!\cdots\!93}{14\!\cdots\!16}a^{10}-\frac{60\!\cdots\!75}{14\!\cdots\!16}a^{9}+\frac{91\!\cdots\!11}{31890075924564}a^{8}+\frac{55\!\cdots\!95}{3865463748432}a^{7}-\frac{46\!\cdots\!45}{483182968554}a^{6}-\frac{12\!\cdots\!55}{644243958072}a^{5}+\frac{39\!\cdots\!27}{29283816276}a^{4}+\frac{24\!\cdots\!24}{2440318023}a^{3}-\frac{14\!\cdots\!39}{1996623837}a^{2}-\frac{19\!\cdots\!21}{3993247674}a+\frac{66\!\cdots\!06}{153586449}$ 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:  \( 328442425589000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 328442425589000000 \cdot 1}{2\cdot\sqrt{111534324540107369492059712937081339641856}}\cr\approx \mathstrut & 2.01411810022744 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608)
 
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 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608, 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 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608);
 
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 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608);
 
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 ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.12.24.155$x^{12} - 4 x^{10} + 30 x^{8} + 40 x^{7} + 40 x^{6} + 16 x^{5} + 68 x^{4} - 80 x^{3} + 144 x^{2} + 224 x + 344$$4$$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$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.7.2$x^{8} + 6$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
11.11.20.8$x^{11} + 110 x^{10} + 1221$$11$$1$$20$$C_{11}$$[2]$
\(197\) Copy content Toggle raw display 197.2.0.1$x^{2} + 192 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} + 192 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.2.1$x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$