Normalized defining polynomial
\( x^{12} - 2 x^{11} - 3 x^{10} + 4 x^{9} + 25 x^{8} + 12 x^{7} + 9 x^{6} - 32 x^{5} - 43 x^{4} + 22 x^{3} + 166 x^{2} + 88 x + 289 \)
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: | \(11057373810786304\) \(\medspace = 2^{18}\cdot 59^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.73\) | 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: | $2^{3/2}59^{1/2}\approx 21.72556098240043$ | ||
Ramified primes: | \(2\), \(59\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{6}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{2}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{1668663403608}a^{11}+\frac{36055368435}{556221134536}a^{10}-\frac{12063895940}{208582925451}a^{9}-\frac{67931431345}{417165850902}a^{8}+\frac{435499828021}{1668663403608}a^{7}-\frac{248068161895}{556221134536}a^{6}+\frac{107569513643}{278110567268}a^{5}-\frac{48927500843}{278110567268}a^{4}-\frac{487961840521}{1668663403608}a^{3}+\frac{123941240209}{556221134536}a^{2}+\frac{98272005863}{1668663403608}a+\frac{664050400285}{1668663403608}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$
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: | $\frac{31382702}{69527641817}a^{11}-\frac{289507857}{69527641817}a^{10}+\frac{66787380}{69527641817}a^{9}+\frac{2089480017}{69527641817}a^{8}-\frac{1968424432}{69527641817}a^{7}-\frac{6380121957}{69527641817}a^{6}-\frac{1863420952}{69527641817}a^{5}+\frac{8507767444}{69527641817}a^{4}-\frac{23328139406}{69527641817}a^{3}-\frac{4716868422}{69527641817}a^{2}-\frac{9517978024}{69527641817}a-\frac{22563053306}{69527641817}$, $\frac{706567447}{1668663403608}a^{11}-\frac{17423492873}{1668663403608}a^{10}+\frac{1098835389}{69527641817}a^{9}+\frac{7577947031}{139055283634}a^{8}-\frac{107014881821}{1668663403608}a^{7}-\frac{570505531771}{1668663403608}a^{6}+\frac{43701930707}{834331701804}a^{5}+\frac{390737607857}{834331701804}a^{4}+\frac{142089138241}{1668663403608}a^{3}-\frac{348870776195}{1668663403608}a^{2}-\frac{513961012063}{1668663403608}a-\frac{2871067493653}{1668663403608}$, $\frac{12852288851}{1668663403608}a^{11}-\frac{28233582901}{1668663403608}a^{10}-\frac{6214808605}{208582925451}a^{9}+\frac{8539739063}{139055283634}a^{8}+\frac{125465974061}{556221134536}a^{7}-\frac{20893517405}{556221134536}a^{6}-\frac{226793927393}{834331701804}a^{5}-\frac{50933109093}{278110567268}a^{4}+\frac{683594521949}{1668663403608}a^{3}+\frac{1894891193209}{1668663403608}a^{2}+\frac{762669405047}{556221134536}a-\frac{2325355228369}{1668663403608}$, $\frac{7666673959}{1668663403608}a^{11}-\frac{18599571113}{1668663403608}a^{10}-\frac{722404629}{69527641817}a^{9}+\frac{4736743251}{139055283634}a^{8}+\frac{143877964363}{1668663403608}a^{7}-\frac{24214774747}{1668663403608}a^{6}+\frac{20816484923}{834331701804}a^{5}+\frac{161183595185}{834331701804}a^{4}-\frac{524190118679}{1668663403608}a^{3}-\frac{228663832163}{1668663403608}a^{2}+\frac{98551514993}{1668663403608}a+\frac{3323996421707}{1668663403608}$, $\frac{2958609791}{208582925451}a^{11}-\frac{6755867780}{208582925451}a^{10}-\frac{11393749877}{208582925451}a^{9}+\frac{23492558531}{208582925451}a^{8}+\frac{83628622549}{208582925451}a^{7}-\frac{6491074006}{69527641817}a^{6}-\frac{73138786868}{208582925451}a^{5}-\frac{25873381391}{69527641817}a^{4}-\frac{44220159977}{69527641817}a^{3}-\frac{27217677808}{208582925451}a^{2}+\frac{445565505857}{208582925451}a+\frac{2790230702}{208582925451}$ | 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: | \( 1147.47282899 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1147.47282899 \cdot 4}{2\cdot\sqrt{11057373810786304}}\cr\approx \mathstrut & 1.34284417797 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T24):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
\(\Q(\sqrt{-2}) \), 3.1.59.1, 6.2.1643032.1, 6.0.1782272.1, 6.0.13144256.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.1643032.1, 6.0.27848.1 |
Degree 8 siblings: | 8.0.14258176.2, 8.4.49632710656.3 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.27848.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | R |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
\(59\) | 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |