Normalized defining polynomial
\( x^{12} - x^{11} - 16 x^{10} + 9 x^{9} + 90 x^{8} - 16 x^{7} - 225 x^{6} - 26 x^{5} + 249 x^{4} + \cdots - 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(5576647516812869\) \(\medspace = 7^{8}\cdot 23^{3}\cdot 43^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(20.52\) | 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))
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Galois root discriminant: | $7^{2/3}23^{1/2}43^{1/2}\approx 115.07920132818636$ | ||
Ramified primes: | \(7\), \(23\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{989}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{4943}a^{11}-\frac{683}{4943}a^{10}+\frac{1148}{4943}a^{9}-\frac{1933}{4943}a^{8}-\frac{1385}{4943}a^{7}+\frac{441}{4943}a^{6}+\frac{536}{4943}a^{5}+\frac{204}{4943}a^{4}-\frac{475}{4943}a^{3}-\frac{2207}{4943}a^{2}+\frac{2405}{4943}a+\frac{819}{4943}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{9498}{4943}a^{11}-\frac{16747}{4943}a^{10}-\frac{138958}{4943}a^{9}+\frac{191445}{4943}a^{8}+\frac{705442}{4943}a^{7}-\frac{690123}{4943}a^{6}-\frac{1596951}{4943}a^{5}+\frac{973707}{4943}a^{4}+\frac{1607884}{4943}a^{3}-\frac{463465}{4943}a^{2}-\frac{572301}{4943}a-\frac{1420}{4943}$, $\frac{8247}{4943}a^{11}-\frac{12510}{4943}a^{10}-\frac{126807}{4943}a^{9}+\frac{143071}{4943}a^{8}+\frac{683312}{4943}a^{7}-\frac{520136}{4943}a^{6}-\frac{1634783}{4943}a^{5}+\frac{738275}{4943}a^{4}+\frac{1717695}{4943}a^{3}-\frac{342070}{4943}a^{2}-\frac{620099}{4943}a-\frac{22560}{4943}$, $\frac{12566}{4943}a^{11}-\frac{21302}{4943}a^{10}-\frac{185740}{4943}a^{9}+\frac{242031}{4943}a^{8}+\frac{954392}{4943}a^{7}-\frac{864522}{4943}a^{6}-\frac{2181796}{4943}a^{5}+\frac{1204139}{4943}a^{4}+\frac{2201929}{4943}a^{3}-\frac{571377}{4943}a^{2}-\frac{771380}{4943}a+\frac{5171}{4943}$, $\frac{8736}{4943}a^{11}-\frac{15316}{4943}a^{10}-\frac{128937}{4943}a^{9}+\frac{176548}{4943}a^{8}+\frac{663466}{4943}a^{7}-\frac{645554}{4943}a^{6}-\frac{1530855}{4943}a^{5}+\frac{927005}{4943}a^{4}+\frac{1589223}{4943}a^{3}-\frac{442579}{4943}a^{2}-\frac{600716}{4943}a-\frac{22452}{4943}$, $\frac{13590}{4943}a^{11}-\frac{23731}{4943}a^{10}-\frac{201451}{4943}a^{9}+\frac{274440}{4943}a^{8}+\frac{1043767}{4943}a^{7}-\frac{1011041}{4943}a^{6}-\frac{2428755}{4943}a^{5}+\frac{1477294}{4943}a^{4}+\frac{2531124}{4943}a^{3}-\frac{745399}{4943}a^{2}-\frac{933393}{4943}a+\frac{3517}{4943}$, $\frac{6902}{4943}a^{11}-\frac{13273}{4943}a^{10}-\frac{98993}{4943}a^{9}+\frac{152824}{4943}a^{8}+\frac{494792}{4943}a^{7}-\frac{554722}{4943}a^{6}-\frac{1124896}{4943}a^{5}+\frac{790133}{4943}a^{4}+\frac{1165307}{4943}a^{3}-\frac{378999}{4943}a^{2}-\frac{434268}{4943}a-\frac{2054}{4943}$, $\frac{956}{4943}a^{11}-\frac{472}{4943}a^{10}-\frac{14687}{4943}a^{9}+\frac{734}{4943}a^{8}+\frac{74809}{4943}a^{7}+\frac{21213}{4943}a^{6}-\frac{154889}{4943}a^{5}-\frac{66955}{4943}a^{4}+\frac{124231}{4943}a^{3}+\frac{50199}{4943}a^{2}-\frac{24030}{4943}a-\frac{2973}{4943}$, $\frac{8034}{4943}a^{11}-\frac{15321}{4943}a^{10}-\frac{114295}{4943}a^{9}+\frac{174189}{4943}a^{8}+\frac{563105}{4943}a^{7}-\frac{619012}{4943}a^{6}-\frac{1254651}{4943}a^{5}+\frac{857942}{4943}a^{4}+\frac{1270197}{4943}a^{3}-\frac{395937}{4943}a^{2}-\frac{465059}{4943}a-\frac{9173}{4943}$, $\frac{9498}{4943}a^{11}-\frac{16747}{4943}a^{10}-\frac{138958}{4943}a^{9}+\frac{191445}{4943}a^{8}+\frac{705442}{4943}a^{7}-\frac{690123}{4943}a^{6}-\frac{1596951}{4943}a^{5}+\frac{973707}{4943}a^{4}+\frac{1607884}{4943}a^{3}-\frac{463465}{4943}a^{2}-\frac{577244}{4943}a-\frac{1420}{4943}$, $\frac{15399}{4943}a^{11}-\frac{28471}{4943}a^{10}-\frac{220551}{4943}a^{9}+\frac{321774}{4943}a^{8}+\frac{1093833}{4943}a^{7}-\frac{1132670}{4943}a^{6}-\frac{2442788}{4943}a^{5}+\frac{1544807}{4943}a^{4}+\frac{2462729}{4943}a^{3}-\frac{694488}{4943}a^{2}-\frac{893044}{4943}a-\frac{22527}{4943}$, $\frac{5179}{4943}a^{11}-\frac{7955}{4943}a^{10}-\frac{80025}{4943}a^{9}+\frac{92485}{4943}a^{8}+\frac{434362}{4943}a^{7}-\frac{345737}{4943}a^{6}-\frac{1049938}{4943}a^{5}+\frac{507843}{4943}a^{4}+\frac{1123650}{4943}a^{3}-\frac{234158}{4943}a^{2}-\frac{421020}{4943}a-\frac{29151}{4943}$ | 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: | \( 6486.54804926 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 6486.54804926 \cdot 1}{2\cdot\sqrt{5576647516812869}}\cr\approx \mathstrut & 0.177892236147 \end{aligned}\]
Galois group
$C_3\times S_4$ (as 12T45):
A solvable group of order 72 |
The 15 conjugacy class representatives for $C_3\times S_4$ |
Character table for $C_3\times S_4$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 4.4.48461.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 siblings: | data not computed |
Degree 24 siblings: | 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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
\(23\) | 23.6.3.1 | $x^{6} + 1058 x^{2} - 219006$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
23.6.0.1 | $x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(43\) | 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |