Normalized defining polynomial
\( x^{12} + 156x^{10} + 9126x^{8} + 246064x^{6} + 2998905x^{4} + 13366548x^{2} + 4826809 \)
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: |
\(31373472824750103330816\)
\(\medspace = 2^{24}\cdot 3^{18}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(74.94\) | 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: | $2^{2}3^{3/2}13^{1/2}\approx 74.93997598078077$ | ||
Ramified primes: |
\(2\), \(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(936=2^{3}\cdot 3^{2}\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{936}(1,·)$, $\chi_{936}(259,·)$, $\chi_{936}(389,·)$, $\chi_{936}(77,·)$, $\chi_{936}(911,·)$, $\chi_{936}(625,·)$, $\chi_{936}(883,·)$, $\chi_{936}(599,·)$, $\chi_{936}(313,·)$, $\chi_{936}(571,·)$, $\chi_{936}(701,·)$, $\chi_{936}(287,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-78}) \), 6.0.7380232704.12$^{3}$, 6.0.22140698112.9$^{3}$, 12.0.31373472824750103330816.5$^{24}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{13}a^{2}$, $\frac{1}{13}a^{3}$, $\frac{1}{169}a^{4}$, $\frac{1}{169}a^{5}$, $\frac{1}{2197}a^{6}$, $\frac{1}{2197}a^{7}$, $\frac{1}{28561}a^{8}$, $\frac{1}{28561}a^{9}$, $\frac{1}{371293}a^{10}$, $\frac{1}{371293}a^{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{5418}$, which has order $10836$ (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: |
$\frac{1}{28561}a^{8}+\frac{8}{2197}a^{6}+\frac{19}{169}a^{4}+\frac{12}{13}a^{2}$, $\frac{1}{28561}a^{8}+\frac{8}{2197}a^{6}+\frac{19}{169}a^{4}+\frac{12}{13}a^{2}+1$, $\frac{1}{371293}a^{10}+\frac{10}{28561}a^{8}+\frac{35}{2197}a^{6}+\frac{50}{169}a^{4}+\frac{24}{13}a^{2}+1$, $\frac{1}{28561}a^{8}+\frac{8}{2197}a^{6}+\frac{20}{169}a^{4}+\frac{15}{13}a^{2}$, $\frac{1}{371293}a^{10}+\frac{11}{28561}a^{8}+\frac{43}{2197}a^{6}+\frac{69}{169}a^{4}+\frac{36}{13}a^{2}$
| 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: | \( 325.67540279491664 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 325.67540279491664 \cdot 10836}{2\cdot\sqrt{31373472824750103330816}}\cr\approx \mathstrut & 0.612945516169711 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_6$ (as 12T2):
An abelian group of order 12 |
The 12 conjugacy class representatives for $C_6\times C_2$ |
Character table for $C_6\times C_2$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-78}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-26})\), \(\Q(\zeta_{36})^+\), 6.0.7380232704.12, 6.0.22140698112.9 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.12.24.318 | $x^{12} + 10 x^{10} + 12 x^{9} + 110 x^{8} + 80 x^{7} + 752 x^{6} + 512 x^{5} + 1636 x^{4} + 1504 x^{3} + 1224 x^{2} + 1008 x - 648$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
\(3\)
| 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
\(13\)
| 13.6.3.2 | $x^{6} + 338 x^{2} - 24167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
13.6.3.2 | $x^{6} + 338 x^{2} - 24167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |