Normalized defining polynomial
\( x^{12} + 253x^{10} + 23276x^{8} + 949026x^{6} + 16790460x^{4} + 102981488x^{2} + 148035889 \)
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: | \(124863324110553424564224\) \(\medspace = 2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 23^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(84.08\) | 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: | $2\cdot 3^{1/2}7^{5/6}23^{1/2}\approx 84.08197566520651$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(23\) | 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: | \(1932=2^{2}\cdot 3\cdot 7\cdot 23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1932}(367,·)$, $\chi_{1932}(1,·)$, $\chi_{1932}(643,·)$, $\chi_{1932}(1381,·)$, $\chi_{1932}(1195,·)$, $\chi_{1932}(461,·)$, $\chi_{1932}(1103,·)$, $\chi_{1932}(275,·)$, $\chi_{1932}(277,·)$, $\chi_{1932}(185,·)$, $\chi_{1932}(827,·)$, $\chi_{1932}(1013,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{-161}) \), 6.0.13087409216.1$^{3}$, 6.0.50480006976.3$^{3}$, 12.0.124863324110553424564224.6$^{24}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{23}a^{2}$, $\frac{1}{23}a^{3}$, $\frac{1}{529}a^{4}$, $\frac{1}{529}a^{5}$, $\frac{1}{12167}a^{6}$, $\frac{1}{12167}a^{7}$, $\frac{1}{279841}a^{8}$, $\frac{1}{279841}a^{9}$, $\frac{1}{6436343}a^{10}$, $\frac{1}{6436343}a^{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}\times C_{4368}$, which has order $17472$ (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: | $\frac{1}{12167}a^{6}+\frac{6}{529}a^{4}+\frac{9}{23}a^{2}+2$, $\frac{1}{6436343}a^{10}+\frac{10}{279841}a^{8}+\frac{35}{12167}a^{6}+\frac{51}{529}a^{4}+\frac{29}{23}a^{2}+4$, $\frac{1}{529}a^{4}+\frac{4}{23}a^{2}+1$, $\frac{1}{23}a^{2}+1$, $\frac{1}{12167}a^{6}+\frac{6}{529}a^{4}+\frac{8}{23}a^{2}+1$ (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: | \( 140.7987960054707 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 140.7987960054707 \cdot 17472}{2\cdot\sqrt{124863324110553424564224}}\cr\approx \mathstrut & 0.214177246222346 \end{aligned}\] (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
\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-161}) \), \(\Q(\sqrt{-69}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-69})\), \(\Q(\zeta_{21})^+\), 6.0.13087409216.1, 6.0.50480006976.3 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
\(3\) | 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(23\) | 23.12.6.1 | $x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |