Normalized defining polynomial
\( x^{12} + 1047x^{10} + 298395x^{8} + 30228984x^{6} + 840352563x^{4} + 783107838x^{2} + 185472909 \)
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: |
\(27936189811629258922498551985852416\)
\(\medspace = 2^{12}\cdot 3^{6}\cdot 349^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(742.19\) | 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: | $2\cdot 3^{1/2}349^{11/12}\approx 742.1880048730932$ | ||
Ramified primes: |
\(2\), \(3\), \(349\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{349}) \) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(4188=2^{2}\cdot 3\cdot 349\) | ||
Dirichlet character group: | $\lbrace$$\chi_{4188}(1,·)$, $\chi_{4188}(3815,·)$, $\chi_{4188}(2603,·)$, $\chi_{4188}(3961,·)$, $\chi_{4188}(911,·)$, $\chi_{4188}(3863,·)$, $\chi_{4188}(2579,·)$, $\chi_{4188}(887,·)$, $\chi_{4188}(697,·)$, $\chi_{4188}(1273,·)$, $\chi_{4188}(3613,·)$, $\chi_{4188}(925,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | 4.0.6121231056.1$^{2}$, 12.0.27936189811629258922498551985852416.1$^{30}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{81}a^{7}+\frac{1}{27}a^{5}-\frac{1}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{37179}a^{8}+\frac{223}{12393}a^{6}+\frac{89}{4131}a^{4}+\frac{55}{1377}a^{2}+\frac{25}{51}$, $\frac{1}{111537}a^{9}+\frac{223}{37179}a^{7}-\frac{370}{12393}a^{5}+\frac{55}{4131}a^{3}+\frac{76}{153}a$, $\frac{1}{17\!\cdots\!71}a^{10}+\frac{48445668610}{58\!\cdots\!57}a^{8}+\frac{9709297974026}{19\!\cdots\!19}a^{6}-\frac{2127837865094}{654695636274873}a^{4}-\frac{2196158151313}{24247986528699}a^{2}+\frac{298958686747}{898073575137}$, $\frac{1}{53\!\cdots\!13}a^{11}+\frac{48445668610}{17\!\cdots\!71}a^{9}+\frac{9709297974026}{58\!\cdots\!57}a^{7}+\frac{70616121721003}{19\!\cdots\!19}a^{5}-\frac{10278820327546}{72743959586097}a^{3}-\frac{599114888390}{2694220725411}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{31076}$, which has order $31821824$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: |
$\frac{287578684}{58\!\cdots\!57}a^{10}+\frac{100340250172}{19\!\cdots\!19}a^{8}+\frac{9524739867569}{654695636274873}a^{6}+\frac{18870377906564}{12837169338723}a^{4}+\frac{19190815725478}{475450716249}a^{2}+\frac{5520249062278}{299357858379}$, $\frac{284178382}{19\!\cdots\!19}a^{10}+\frac{99193123018}{654695636274873}a^{8}+\frac{9426175050059}{218231878758291}a^{6}+\frac{318344849606530}{72743959586097}a^{4}+\frac{979838803371218}{8082662176233}a^{2}+\frac{17229648178742}{299357858379}$, $\frac{1228121821}{17\!\cdots\!71}a^{10}+\frac{429310996045}{58\!\cdots\!57}a^{8}+\frac{40914663362837}{19\!\cdots\!19}a^{6}+\frac{13\!\cdots\!67}{654695636274873}a^{4}+\frac{14\!\cdots\!14}{24247986528699}a^{2}+\frac{24736704611479}{898073575137}$, $\frac{10871162015}{58\!\cdots\!57}a^{10}+\frac{223074022375}{115534524048507}a^{8}+\frac{21165472566185}{38511508016169}a^{6}+\frac{12\!\cdots\!09}{218231878758291}a^{4}+\frac{12\!\cdots\!10}{8082662176233}a^{2}+\frac{208617715897556}{299357858379}$, $\frac{6425925937}{570218779981341}a^{10}+\frac{131861747840}{11180760391791}a^{8}+\frac{12511957859224}{3726920130597}a^{6}+\frac{71\!\cdots\!29}{21119214073383}a^{4}+\frac{72\!\cdots\!14}{782193113829}a^{2}+\frac{129926197899478}{28970115327}$
| 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: | \( 49949.033153968034 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 49949.033153968034 \cdot 31821824}{2\cdot\sqrt{27936189811629258922498551985852416}}\cr\approx \mathstrut & 0.292561910669375 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 12 |
The 12 conjugacy class representatives for $C_{12}$ |
Character table for $C_{12}$ |
Intermediate fields
\(\Q(\sqrt{349}) \), 3.3.121801.1, 4.0.6121231056.1, 6.6.5177583776749.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.1.0.1}{1} }^{12}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\)
| 2.12.12.25 | $x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(349\)
| Deg $12$ | $12$ | $1$ | $11$ |