Normalized defining polynomial
\( x^{12} - 2x^{11} + 3x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 4x^{6} - 2x^{5} - x^{4} + 2x^{2} - x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(63518590021\) \(\medspace = 23^{4}\cdot 61^{3}\) | 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: | \(7.95\) | 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: | $23^{1/2}61^{3/4}\approx 104.6794167699507$ | ||
Ramified primes: | \(23\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{137}a^{11}-\frac{37}{137}a^{10}+\frac{65}{137}a^{9}+\frac{52}{137}a^{8}-\frac{37}{137}a^{7}+\frac{58}{137}a^{6}+\frac{29}{137}a^{5}-\frac{58}{137}a^{4}-\frac{26}{137}a^{3}-\frac{49}{137}a^{2}-\frac{64}{137}a+\frac{47}{137}$
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)
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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: | $\frac{74}{137}a^{11}-\frac{135}{137}a^{10}+\frac{152}{137}a^{9}-\frac{125}{137}a^{8}+\frac{139}{137}a^{7}-\frac{366}{137}a^{6}+\frac{228}{137}a^{5}-\frac{45}{137}a^{4}-\frac{6}{137}a^{3}-\frac{64}{137}a^{2}+\frac{196}{137}a+\frac{53}{137}$, $\frac{8}{137}a^{11}-\frac{22}{137}a^{10}-\frac{28}{137}a^{9}+\frac{5}{137}a^{8}-\frac{22}{137}a^{7}-\frac{84}{137}a^{6}-\frac{42}{137}a^{5}+\frac{84}{137}a^{4}+\frac{66}{137}a^{3}-\frac{118}{137}a^{2}+\frac{36}{137}a+\frac{102}{137}$, $a$, $\frac{49}{137}a^{11}-\frac{32}{137}a^{10}+\frac{34}{137}a^{9}+\frac{82}{137}a^{8}-\frac{32}{137}a^{7}-\frac{35}{137}a^{6}-\frac{86}{137}a^{5}+\frac{172}{137}a^{4}-\frac{178}{137}a^{3}-\frac{72}{137}a^{2}+\frac{152}{137}a+\frac{111}{137}$, $\frac{50}{137}a^{11}-\frac{69}{137}a^{10}+\frac{99}{137}a^{9}-\frac{3}{137}a^{8}+\frac{68}{137}a^{7}-\frac{114}{137}a^{6}+\frac{80}{137}a^{5}+\frac{114}{137}a^{4}-\frac{204}{137}a^{3}+\frac{16}{137}a^{2}+\frac{88}{137}a+\frac{21}{137}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 1.51948050623 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1.51948050623 \cdot 1}{2\cdot\sqrt{63518590021}}\cr\approx \mathstrut & 0.185478648498 \end{aligned}\]
Galois group
$C_4\wr S_3$ (as 12T150):
A solvable group of order 384 |
The 40 conjugacy class representatives for $C_4\wr S_3$ |
Character table for $C_4\wr S_3$ |
Intermediate fields
3.1.23.1, 6.2.32269.1 |
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 |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | R | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(61\) | 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.4.3.2 | $x^{4} + 61$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |