Normalized defining polynomial
\( x^{12} - 968 x^{10} + 346060 x^{8} - 56221440 x^{6} + 4128059232 x^{4} - 247374336 x^{3} - 114286943232 x^{2} + 19295198208 x + 632319724608 \)
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: | \(111534324540107369492059712937081339641856\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 11^{20}\cdot 197^{4}\) | 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: | \(2\), \(3\), \(11\), \(197\) | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{22}a^{3}$, $\frac{1}{44}a^{4}$, $\frac{1}{484}a^{5}$, $\frac{1}{968}a^{6}$, $\frac{1}{21296}a^{7}-\frac{1}{968}a^{5}-\frac{1}{2}a$, $\frac{1}{276848}a^{8}-\frac{5}{276848}a^{7}-\frac{3}{12584}a^{6}+\frac{1}{1144}a^{5}-\frac{1}{572}a^{4}+\frac{5}{286}a^{3}+\frac{3}{26}a^{2}+\frac{1}{26}a$, $\frac{1}{6090656}a^{9}+\frac{1}{12584}a^{6}-\frac{5}{12584}a^{5}+\frac{1}{286}a^{4}-\frac{3}{572}a^{3}-\frac{2}{13}a^{2}-\frac{11}{26}a$, $\frac{1}{103541152}a^{10}+\frac{1}{12942644}a^{9}-\frac{1}{2353208}a^{8}-\frac{5}{276848}a^{7}+\frac{6}{26741}a^{6}-\frac{109}{213928}a^{5}-\frac{3}{286}a^{4}-\frac{46}{2431}a^{3}+\frac{21}{221}a^{2}-\frac{3}{26}a-\frac{5}{17}$, $\frac{1}{61\!\cdots\!04}a^{11}-\frac{6112003}{28\!\cdots\!32}a^{10}+\frac{72331165}{14\!\cdots\!16}a^{9}-\frac{33810641}{127560303698256}a^{8}-\frac{12724963}{3270777017904}a^{7}+\frac{178788053}{966365937108}a^{6}-\frac{153608237}{644243958072}a^{5}+\frac{6829369}{2662165116}a^{4}+\frac{90450011}{9761272092}a^{3}-\frac{499045985}{1996623837}a^{2}-\frac{114366082}{1996623837}a+\frac{22247857}{153586449}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{6774911265461}{25\!\cdots\!96}a^{11}-\frac{43786837078279}{935442227120544}a^{10}-\frac{803745684348251}{467721113560272}a^{9}+\frac{12\!\cdots\!43}{42520101232752}a^{8}+\frac{26\!\cdots\!27}{7086683538792}a^{7}-\frac{192307474288877}{29283816276}a^{6}-\frac{846553275838714}{26843498253}a^{5}+\frac{13\!\cdots\!42}{2440318023}a^{4}+\frac{737641189453826}{813439341}a^{3}-\frac{22\!\cdots\!29}{1331082558}a^{2}-\frac{122396976163219}{51195483}a+\frac{48\!\cdots\!44}{51195483}$, $\frac{17182472009}{36\!\cdots\!12}a^{11}-\frac{3183853}{288597972168}a^{10}-\frac{486879121799}{165078040080096}a^{9}+\frac{4673070067}{937943409546}a^{8}+\frac{1352446416697}{2501182425456}a^{7}-\frac{108012053329}{113690110248}a^{6}-\frac{11138312707}{430644357}a^{5}+\frac{53202400051}{430644357}a^{4}-\frac{11275781756}{11042163}a^{3}-\frac{1001275578715}{117448461}a^{2}+\frac{1098424199501}{18068994}a+\frac{1688229794057}{9034497}$, $\frac{39891587461373}{61\!\cdots\!04}a^{11}-\frac{19083084691499}{28\!\cdots\!32}a^{10}-\frac{15\!\cdots\!83}{28\!\cdots\!32}a^{9}+\frac{67053840375835}{11596391245296}a^{8}+\frac{68\!\cdots\!85}{42520101232752}a^{7}-\frac{32\!\cdots\!21}{1932731874216}a^{6}-\frac{14\!\cdots\!09}{80530494759}a^{5}+\frac{54\!\cdots\!45}{29283816276}a^{4}+\frac{31\!\cdots\!93}{4880636046}a^{3}-\frac{13\!\cdots\!80}{1996623837}a^{2}-\frac{16\!\cdots\!09}{3993247674}a+\frac{63\!\cdots\!61}{153586449}$, $\frac{75394044722291}{68\!\cdots\!56}a^{11}-\frac{5458496956885}{28346734155168}a^{10}-\frac{171977679413381}{23985698131296}a^{9}+\frac{17\!\cdots\!15}{14173367077584}a^{8}+\frac{73\!\cdots\!01}{4724455692528}a^{7}-\frac{28\!\cdots\!77}{107373993012}a^{6}-\frac{49\!\cdots\!47}{35791331004}a^{5}+\frac{36\!\cdots\!81}{1626878682}a^{4}+\frac{24\!\cdots\!93}{542292894}a^{3}-\frac{29\!\cdots\!69}{443694186}a^{2}-\frac{12\!\cdots\!41}{443694186}a+\frac{56\!\cdots\!91}{17065161}$, $\frac{112687924600997}{15\!\cdots\!76}a^{11}-\frac{92004429225209}{701581670340408}a^{10}-\frac{13\!\cdots\!93}{28\!\cdots\!32}a^{9}+\frac{980599381496737}{11596391245296}a^{8}+\frac{26\!\cdots\!35}{2657506327047}a^{7}-\frac{34\!\cdots\!93}{1932731874216}a^{6}-\frac{55\!\cdots\!09}{644243958072}a^{5}+\frac{11\!\cdots\!61}{7320954069}a^{4}+\frac{24\!\cdots\!79}{9761272092}a^{3}-\frac{14\!\cdots\!59}{307172898}a^{2}-\frac{26\!\cdots\!83}{3993247674}a+\frac{39\!\cdots\!19}{153586449}$, $\frac{18\!\cdots\!75}{15\!\cdots\!76}a^{11}-\frac{17\!\cdots\!11}{14\!\cdots\!16}a^{10}-\frac{12\!\cdots\!95}{127560303698256}a^{9}+\frac{13\!\cdots\!59}{127560303698256}a^{8}+\frac{94\!\cdots\!07}{3270777017904}a^{7}-\frac{30\!\cdots\!61}{966365937108}a^{6}-\frac{20\!\cdots\!55}{644243958072}a^{5}+\frac{25\!\cdots\!01}{7320954069}a^{4}+\frac{26\!\cdots\!51}{2440318023}a^{3}-\frac{23\!\cdots\!41}{1996623837}a^{2}-\frac{16\!\cdots\!03}{3993247674}a+\frac{10\!\cdots\!05}{153586449}$, $\frac{285292839131}{30\!\cdots\!52}a^{11}+\frac{266151168175}{14\!\cdots\!16}a^{10}-\frac{9401127398383}{28\!\cdots\!32}a^{9}-\frac{1475970593639}{9812331053712}a^{8}-\frac{10546622743351}{10630025308188}a^{7}+\frac{70087782859867}{1932731874216}a^{6}+\frac{328976359869589}{644243958072}a^{5}-\frac{70085640844513}{29283816276}a^{4}-\frac{533259898978855}{9761272092}a^{3}-\frac{60225385311290}{1996623837}a^{2}+\frac{60\!\cdots\!79}{3993247674}a+\frac{508483517324281}{153586449}$, $\frac{148068286815589}{68\!\cdots\!56}a^{11}+\frac{353875741435}{2180518011936}a^{10}-\frac{60\!\cdots\!39}{311814075706848}a^{9}-\frac{20\!\cdots\!07}{14173367077584}a^{8}+\frac{29\!\cdots\!93}{4724455692528}a^{7}+\frac{98\!\cdots\!29}{214747986024}a^{6}-\frac{15\!\cdots\!27}{17895665502}a^{5}-\frac{17\!\cdots\!35}{295796124}a^{4}+\frac{11\!\cdots\!36}{271146447}a^{3}+\frac{66\!\cdots\!47}{221847093}a^{2}-\frac{11\!\cdots\!61}{34130322}a-\frac{32\!\cdots\!74}{17065161}$, $\frac{15\!\cdots\!39}{61\!\cdots\!04}a^{11}+\frac{11\!\cdots\!87}{701581670340408}a^{10}-\frac{63\!\cdots\!25}{28\!\cdots\!32}a^{9}-\frac{94\!\cdots\!99}{63780151849128}a^{8}+\frac{15\!\cdots\!37}{21260050616376}a^{7}+\frac{11\!\cdots\!68}{241591484277}a^{6}-\frac{67\!\cdots\!67}{644243958072}a^{5}-\frac{20\!\cdots\!73}{29283816276}a^{4}+\frac{26\!\cdots\!11}{4880636046}a^{3}+\frac{14\!\cdots\!75}{3993247674}a^{2}-\frac{84\!\cdots\!15}{1996623837}a-\frac{35\!\cdots\!09}{153586449}$, $\frac{45\!\cdots\!93}{30\!\cdots\!52}a^{11}+\frac{76\!\cdots\!19}{14\!\cdots\!16}a^{10}-\frac{39\!\cdots\!63}{28\!\cdots\!32}a^{9}-\frac{65\!\cdots\!29}{127560303698256}a^{8}+\frac{20\!\cdots\!73}{42520101232752}a^{7}+\frac{17\!\cdots\!91}{966365937108}a^{6}-\frac{10\!\cdots\!67}{14641908138}a^{5}-\frac{38\!\cdots\!93}{14641908138}a^{4}+\frac{36\!\cdots\!55}{750867084}a^{3}+\frac{32\!\cdots\!78}{1996623837}a^{2}-\frac{20\!\cdots\!64}{1996623837}a-\frac{43\!\cdots\!77}{153586449}$, $\frac{14\!\cdots\!39}{30\!\cdots\!52}a^{11}-\frac{42\!\cdots\!93}{14\!\cdots\!16}a^{10}-\frac{60\!\cdots\!75}{14\!\cdots\!16}a^{9}+\frac{91\!\cdots\!11}{31890075924564}a^{8}+\frac{55\!\cdots\!95}{3865463748432}a^{7}-\frac{46\!\cdots\!45}{483182968554}a^{6}-\frac{12\!\cdots\!55}{644243958072}a^{5}+\frac{39\!\cdots\!27}{29283816276}a^{4}+\frac{24\!\cdots\!24}{2440318023}a^{3}-\frac{14\!\cdots\!39}{1996623837}a^{2}-\frac{19\!\cdots\!21}{3993247674}a+\frac{66\!\cdots\!06}{153586449}$ (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)
Galois group
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 $\Q$. |
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.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 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 | $[2, 2, 2, 3, 3]^{3}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | 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 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.11.20.8 | $x^{11} + 110 x^{10} + 1221$ | $11$ | $1$ | $20$ | $C_{11}$ | $[2]$ | |
\(197\) | 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 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |