Normalized defining polynomial
\( x^{12} - 4x^{11} + 8x^{10} - 10x^{9} + 8x^{8} - 3x^{7} - 2x^{6} + 4x^{5} - 2x^{4} - x^{3} + 2x^{2} - x + 1 \)
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: | \(46779109213\) \(\medspace = 31^{4}\cdot 37^{3}\) | 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: | \(7.75\) | 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: | $31^{1/2}37^{3/4}\approx 83.52801100380672$ | ||
Ramified primes: | \(31\), \(37\) | 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: | \(\Q(\sqrt{37}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17}a^{11}+\frac{8}{17}a^{10}+\frac{2}{17}a^{9}-\frac{3}{17}a^{8}+\frac{6}{17}a^{7}+\frac{1}{17}a^{6}-\frac{7}{17}a^{5}+\frac{5}{17}a^{4}+\frac{7}{17}a^{3}-\frac{2}{17}a^{2}-\frac{5}{17}a+\frac{7}{17}$
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: | $\frac{6}{17}a^{11}-\frac{20}{17}a^{10}+\frac{46}{17}a^{9}-\frac{69}{17}a^{8}+\frac{70}{17}a^{7}-\frac{45}{17}a^{6}+\frac{9}{17}a^{5}+\frac{13}{17}a^{4}-\frac{9}{17}a^{3}-\frac{12}{17}a^{2}+\frac{21}{17}a-\frac{9}{17}$, $\frac{6}{17}a^{11}-\frac{20}{17}a^{10}+\frac{29}{17}a^{9}-\frac{18}{17}a^{8}-\frac{15}{17}a^{7}+\frac{40}{17}a^{6}-\frac{42}{17}a^{5}+\frac{13}{17}a^{4}+\frac{25}{17}a^{3}-\frac{29}{17}a^{2}+\frac{4}{17}a+\frac{8}{17}$, $\frac{4}{17}a^{11}-\frac{19}{17}a^{10}+\frac{42}{17}a^{9}-\frac{46}{17}a^{8}+\frac{24}{17}a^{7}+\frac{4}{17}a^{6}-\frac{11}{17}a^{5}+\frac{3}{17}a^{4}+\frac{11}{17}a^{3}-\frac{25}{17}a^{2}+\frac{14}{17}a+\frac{11}{17}$, $\frac{3}{17}a^{11}-\frac{10}{17}a^{10}+\frac{23}{17}a^{9}-\frac{43}{17}a^{8}+\frac{52}{17}a^{7}-\frac{31}{17}a^{6}-\frac{4}{17}a^{5}+\frac{32}{17}a^{4}-\frac{30}{17}a^{3}+\frac{11}{17}a^{2}+\frac{2}{17}a-\frac{13}{17}$, $\frac{6}{17}a^{11}-\frac{20}{17}a^{10}+\frac{29}{17}a^{9}-\frac{18}{17}a^{8}-\frac{15}{17}a^{7}+\frac{40}{17}a^{6}-\frac{42}{17}a^{5}+\frac{30}{17}a^{4}-\frac{9}{17}a^{3}-\frac{12}{17}a^{2}+\frac{4}{17}a+\frac{8}{17}$ | 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: | \( 1.25722954293 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1.25722954293 \cdot 1}{2\cdot\sqrt{46779109213}}\cr\approx \mathstrut & 0.178828946640 \end{aligned}\]
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 $C_4\wr S_3$ is not computed |
Intermediate fields
3.1.31.1, 6.2.35557.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | R | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(31\) | 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(37\) | 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.4.3.2 | $x^{4} + 37$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |