Normalized defining polynomial
\( x^{12} - x^{10} - 2x^{9} - 4x^{8} + 8x^{7} - 10x^{6} + 4x^{5} + 38x^{4} - 64x^{3} + 50x^{2} - 20x + 4 \)
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: |
\(47464824438784\)
\(\medspace = 2^{26}\cdot 29^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.79\) | 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: | $2^{23/8}29^{1/2}\approx 39.50574321172675$ | ||
Ramified primes: |
\(2\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}$, $\frac{1}{286658}a^{11}+\frac{59531}{286658}a^{10}-\frac{6447}{143329}a^{9}+\frac{38704}{143329}a^{8}-\frac{69182}{143329}a^{7}-\frac{58152}{143329}a^{6}-\frac{21380}{143329}a^{5}-\frac{11258}{143329}a^{4}+\frac{6425}{143329}a^{3}-\frac{58458}{143329}a^{2}-\frac{35053}{143329}a-\frac{13242}{143329}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
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: |
$\frac{54558}{143329}a^{11}-\frac{29013}{286658}a^{10}-\frac{167569}{286658}a^{9}-\frac{106650}{143329}a^{8}-\frac{154669}{143329}a^{7}+\frac{577843}{143329}a^{6}-\frac{507263}{143329}a^{5}+\frac{188260}{143329}a^{4}+\frac{2198096}{143329}a^{3}-\frac{4002524}{143329}a^{2}+\frac{2471139}{143329}a-\frac{587739}{143329}$, $\frac{10594}{143329}a^{11}+\frac{23814}{143329}a^{10}-\frac{6499}{143329}a^{9}-\frac{68186}{143329}a^{8}-\frac{145862}{143329}a^{7}-\frac{68492}{143329}a^{6}+\frac{63529}{143329}a^{5}-\frac{35048}{143329}a^{4}+\frac{543666}{143329}a^{3}+\frac{471101}{143329}a^{2}-\frac{402073}{143329}a+\frac{210015}{143329}$, $\frac{98930}{143329}a^{11}+\frac{13220}{143329}a^{10}-\frac{118649}{143329}a^{9}-\frac{238359}{143329}a^{8}-\frac{431020}{143329}a^{7}+\frac{784058}{143329}a^{6}-\frac{751339}{143329}a^{5}+\frac{254767}{143329}a^{4}+\frac{3935482}{143329}a^{3}-\frac{5726069}{143329}a^{2}+\frac{3400297}{143329}a-\frac{1011303}{143329}$, $\frac{57131}{286658}a^{11}+\frac{5860}{143329}a^{10}-\frac{79383}{286658}a^{9}-\frac{81588}{143329}a^{8}-\frac{139667}{143329}a^{7}+\frac{227637}{143329}a^{6}-\frac{154371}{143329}a^{5}+\frac{79754}{143329}a^{4}+\frac{1147738}{143329}a^{3}-\frac{1631588}{143329}a^{2}+\frac{839819}{143329}a-\frac{181569}{143329}$, $\frac{148527}{286658}a^{11}-\frac{5173}{286658}a^{10}-\frac{115849}{143329}a^{9}-\frac{193853}{143329}a^{8}-\frac{282233}{143329}a^{7}+\frac{723410}{143329}a^{6}-\frac{483252}{143329}a^{5}+\frac{245806}{143329}a^{4}+\frac{3011402}{143329}a^{3}-\frac{4737061}{143329}a^{2}+\frac{2402258}{143329}a-\frac{750641}{143329}$
| 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: | \( 172.555606426 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 172.555606426 \cdot 2}{2\cdot\sqrt{47464824438784}}\cr\approx \mathstrut & 1.54106992521 \end{aligned}\]
Galois group
$C_2^2\wr S_3$ (as 12T139):
A solvable group of order 384 |
The 40 conjugacy class representatives for $C_2^2\wr S_3$ |
Character table for $C_2^2\wr S_3$ |
Intermediate fields
3.1.116.1, 6.0.861184.2, 6.2.1722368.1, 6.0.430592.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Minimal sibling: | 12.0.47464824438784.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.8.22.108 | $x^{8} + 4 x^{7} + 14 x^{4} + 12 x^{2} + 14$ | $8$ | $1$ | $22$ | $C_2^2 \wr C_2$ | $[2, 2, 3, 7/2]^{2}$ | |
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |