Normalized defining polynomial
\( x^{12} + 2x^{10} + 7x^{8} - 2x^{6} + 19 \)
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)
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Discriminant: | \(2658691056664576\) \(\medspace = 2^{30}\cdot 19^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.29\) | 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: | $2^{215/64}19^{5/6}\approx 119.37149761252084$ | ||
Ramified primes: | \(2\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2247}a^{10}-\frac{2}{321}a^{8}+\frac{11}{107}a^{6}+\frac{796}{2247}a^{4}+\frac{746}{2247}a^{2}-\frac{701}{2247}$, $\frac{1}{2247}a^{11}-\frac{2}{321}a^{9}+\frac{11}{107}a^{7}+\frac{796}{2247}a^{5}+\frac{746}{2247}a^{3}-\frac{701}{2247}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{131}{2247}a^{10}+\frac{59}{321}a^{8}+\frac{50}{107}a^{6}+\frac{914}{2247}a^{4}-\frac{1142}{2247}a^{2}+\frac{2543}{2247}$, $\frac{17}{2247}a^{10}-\frac{34}{321}a^{8}-\frac{27}{107}a^{6}-\frac{2197}{2247}a^{4}-\frac{800}{2247}a^{2}-\frac{682}{2247}$, $\frac{148}{2247}a^{11}-\frac{53}{749}a^{10}+\frac{25}{321}a^{9}-\frac{1}{107}a^{8}+\frac{23}{107}a^{7}-\frac{37}{107}a^{6}-\frac{1283}{2247}a^{5}+\frac{505}{749}a^{4}-\frac{1942}{2247}a^{3}-\frac{590}{749}a^{2}+\frac{1861}{2247}a-\frac{297}{749}$, $\frac{47}{749}a^{11}+\frac{8}{2247}a^{10}+\frac{13}{107}a^{9}-\frac{16}{321}a^{8}+\frac{53}{107}a^{7}-\frac{19}{107}a^{6}-\frac{38}{749}a^{5}-\frac{373}{2247}a^{4}-\frac{141}{749}a^{3}+\frac{1474}{2247}a^{2}-\frac{740}{749}a+\frac{3380}{2247}$, $\frac{121}{2247}a^{11}+\frac{13}{2247}a^{10}+\frac{79}{321}a^{9}-\frac{26}{321}a^{8}+\frac{47}{107}a^{7}+\frac{36}{107}a^{6}-\frac{305}{2247}a^{5}-\frac{887}{2247}a^{4}-\frac{1861}{2247}a^{3}-\frac{1537}{2247}a^{2}-\frac{1682}{2247}a-\frac{4619}{2247}$ | 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: | \( 1021.08997934 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1021.08997934 \cdot 1}{2\cdot\sqrt{2658691056664576}}\cr\approx \mathstrut & 0.609227208980 \end{aligned}\]
Galois group
$A_4^2:D_4$ (as 12T208):
A solvable group of order 1152 |
The 44 conjugacy class representatives for $A_4^2:D_4$ |
Character table for $A_4^2:D_4$ is not computed |
Intermediate fields
\(\Q(\sqrt{-2}) \), 6.0.184832.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Degree 36 siblings: | 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 | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.30.170 | $x^{12} + 128 x^{9} - 50 x^{8} - 1888 x^{7} - 2144 x^{6} - 1536 x^{5} - 548 x^{4} - 768 x^{3} - 1024 x^{2} + 712$ | $4$ | $3$ | $30$ | 12T134 | $[2, 2, 2, 3, 7/2, 7/2, 7/2]^{3}$ |
\(19\) | 19.6.5.4 | $x^{6} + 76$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |