Normalized defining polynomial
\( x^{12} - x^{11} - 5 x^{10} - 19 x^{9} + 38 x^{8} + 103 x^{7} - 52 x^{6} - 404 x^{5} - 80 x^{4} + \cdots + 4489 \)
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: | \(1572266908616041\) \(\medspace = 11^{6}\cdot 31^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.47\) | 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: | $11^{1/2}31^{1/2}\approx 18.466185312619388$ | ||
Ramified primes: | \(11\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13}a^{10}-\frac{6}{13}a^{9}-\frac{2}{13}a^{8}-\frac{3}{13}a^{7}+\frac{3}{13}a^{6}-\frac{3}{13}a^{4}+\frac{1}{13}a^{3}-\frac{4}{13}a^{2}+\frac{4}{13}a+\frac{4}{13}$, $\frac{1}{43\!\cdots\!83}a^{11}-\frac{44\!\cdots\!84}{14\!\cdots\!61}a^{10}+\frac{90\!\cdots\!60}{43\!\cdots\!83}a^{9}+\frac{15\!\cdots\!76}{48\!\cdots\!87}a^{8}-\frac{10\!\cdots\!98}{43\!\cdots\!83}a^{7}+\frac{55\!\cdots\!68}{14\!\cdots\!61}a^{6}+\frac{11\!\cdots\!45}{43\!\cdots\!83}a^{5}+\frac{23\!\cdots\!77}{14\!\cdots\!61}a^{4}-\frac{12\!\cdots\!23}{43\!\cdots\!83}a^{3}-\frac{66\!\cdots\!99}{43\!\cdots\!83}a^{2}-\frac{19\!\cdots\!86}{43\!\cdots\!83}a+\frac{60\!\cdots\!33}{65\!\cdots\!49}$
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: | $\frac{248635990917818}{43\!\cdots\!83}a^{11}-\frac{610448972840569}{14\!\cdots\!61}a^{10}-\frac{620093819688685}{43\!\cdots\!83}a^{9}+\frac{516644216049914}{48\!\cdots\!87}a^{8}+\frac{35\!\cdots\!04}{43\!\cdots\!83}a^{7}+\frac{232280969394467}{14\!\cdots\!61}a^{6}-\frac{23\!\cdots\!94}{43\!\cdots\!83}a^{5}-\frac{33\!\cdots\!62}{14\!\cdots\!61}a^{4}+\frac{20\!\cdots\!88}{43\!\cdots\!83}a^{3}+\frac{11\!\cdots\!07}{43\!\cdots\!83}a^{2}-\frac{77\!\cdots\!34}{43\!\cdots\!83}a-\frac{12\!\cdots\!73}{65\!\cdots\!49}$, $\frac{467271757930555}{43\!\cdots\!83}a^{11}-\frac{455506500905243}{14\!\cdots\!61}a^{10}-\frac{103453133229686}{43\!\cdots\!83}a^{9}-\frac{754636969603331}{48\!\cdots\!87}a^{8}+\frac{35\!\cdots\!97}{43\!\cdots\!83}a^{7}-\frac{15\!\cdots\!34}{14\!\cdots\!61}a^{6}-\frac{11\!\cdots\!89}{43\!\cdots\!83}a^{5}-\frac{34\!\cdots\!31}{14\!\cdots\!61}a^{4}+\frac{94\!\cdots\!00}{43\!\cdots\!83}a^{3}+\frac{18\!\cdots\!56}{43\!\cdots\!83}a^{2}-\frac{18\!\cdots\!38}{43\!\cdots\!83}a-\frac{384241757102452}{65\!\cdots\!49}$, $\frac{19\!\cdots\!32}{43\!\cdots\!83}a^{11}-\frac{229570258547833}{14\!\cdots\!61}a^{10}-\frac{13\!\cdots\!19}{43\!\cdots\!83}a^{9}-\frac{349367923460047}{37\!\cdots\!99}a^{8}+\frac{11\!\cdots\!40}{43\!\cdots\!83}a^{7}+\frac{99\!\cdots\!87}{14\!\cdots\!61}a^{6}+\frac{25\!\cdots\!16}{43\!\cdots\!83}a^{5}-\frac{62\!\cdots\!35}{14\!\cdots\!61}a^{4}-\frac{71\!\cdots\!16}{43\!\cdots\!83}a^{3}-\frac{12\!\cdots\!39}{43\!\cdots\!83}a^{2}-\frac{11\!\cdots\!03}{43\!\cdots\!83}a-\frac{10\!\cdots\!24}{65\!\cdots\!49}$, $\frac{22\!\cdots\!36}{43\!\cdots\!83}a^{11}+\frac{11\!\cdots\!65}{14\!\cdots\!61}a^{10}-\frac{21\!\cdots\!53}{43\!\cdots\!83}a^{9}-\frac{69\!\cdots\!85}{48\!\cdots\!87}a^{8}-\frac{46\!\cdots\!14}{43\!\cdots\!83}a^{7}+\frac{20\!\cdots\!96}{14\!\cdots\!61}a^{6}+\frac{24\!\cdots\!33}{43\!\cdots\!83}a^{5}-\frac{25\!\cdots\!47}{14\!\cdots\!61}a^{4}-\frac{43\!\cdots\!63}{43\!\cdots\!83}a^{3}+\frac{47\!\cdots\!78}{43\!\cdots\!83}a^{2}+\frac{82\!\cdots\!85}{43\!\cdots\!83}a+\frac{61\!\cdots\!85}{65\!\cdots\!49}$, $\frac{89\!\cdots\!20}{43\!\cdots\!83}a^{11}+\frac{19\!\cdots\!53}{14\!\cdots\!61}a^{10}-\frac{46\!\cdots\!64}{43\!\cdots\!83}a^{9}-\frac{27\!\cdots\!26}{48\!\cdots\!87}a^{8}+\frac{27\!\cdots\!21}{43\!\cdots\!83}a^{7}+\frac{43\!\cdots\!85}{14\!\cdots\!61}a^{6}+\frac{14\!\cdots\!79}{43\!\cdots\!83}a^{5}-\frac{12\!\cdots\!90}{14\!\cdots\!61}a^{4}-\frac{85\!\cdots\!63}{43\!\cdots\!83}a^{3}+\frac{76\!\cdots\!26}{43\!\cdots\!83}a^{2}+\frac{21\!\cdots\!97}{43\!\cdots\!83}a+\frac{66\!\cdots\!03}{65\!\cdots\!49}$ | 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: | \( 123.51771784 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 123.51771784 \cdot 4}{2\cdot\sqrt{1572266908616041}}\cr\approx \mathstrut & 0.38333222302 \end{aligned}\]
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
\(\Q(\sqrt{341}) \), 3.1.31.1, 6.0.10571.1, 6.2.39651821.1, 6.0.3604711.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.327701.1, 6.0.10571.1 |
Degree 8 siblings: | 8.4.13521270961.1, 8.0.14070001.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.10571.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(31\) | 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |