Normalized defining polynomial
\( x^{12} - x^{11} + 2x^{9} - x^{8} - 4x^{7} + 5x^{6} - x^{5} - x^{4} - x^{3} + 4x^{2} - 3x + 1 \)
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: | \(63995902381\) \(\medspace = 73^{2}\cdot 229^{3}\) | 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: | $73^{2/3}229^{1/2}\approx 264.32179884736155$ | ||
Ramified primes: | \(73\), \(229\) | 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: | \(\Q(\sqrt{229}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{11}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}$
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: | $\frac{4}{5}a^{11}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{6}{5}a^{8}+a^{7}-\frac{12}{5}a^{6}-\frac{2}{5}a^{5}+\frac{3}{5}a^{4}+\frac{3}{5}a^{3}-\frac{1}{5}a^{2}+a+\frac{3}{5}$, $a$, $\frac{2}{5}a^{11}-\frac{2}{5}a^{9}+\frac{3}{5}a^{8}+\frac{1}{5}a^{7}-2a^{6}+\frac{3}{5}a^{5}+\frac{6}{5}a^{4}-\frac{1}{5}a^{3}+\frac{7}{5}a-\frac{4}{5}$, $\frac{4}{5}a^{11}-\frac{2}{5}a^{10}-\frac{4}{5}a^{9}+\frac{6}{5}a^{8}-\frac{1}{5}a^{7}-\frac{21}{5}a^{6}+\frac{8}{5}a^{5}+\frac{8}{5}a^{4}-\frac{1}{5}a^{3}-\frac{3}{5}a^{2}+2a-\frac{3}{5}$, $\frac{4}{5}a^{11}+\frac{3}{5}a^{10}-\frac{2}{5}a^{9}+\frac{6}{5}a^{8}+\frac{9}{5}a^{7}-\frac{13}{5}a^{6}-\frac{6}{5}a^{5}+\frac{11}{5}a^{4}+\frac{3}{5}a^{3}-\frac{4}{5}a^{2}+\frac{3}{5}a+\frac{6}{5}$ | 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: | \( 1.54183330651 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1.54183330651 \cdot 1}{2\cdot\sqrt{63995902381}}\cr\approx \mathstrut & 0.187504007321 \end{aligned}\]
Galois group
$C_3\wr S_4$ (as 12T231):
A solvable group of order 1944 |
The 51 conjugacy class representatives for $C_3\wr S_4$ are not computed |
Character table for $C_3\wr S_4$ is not computed |
Intermediate fields
4.0.229.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 sibling: | 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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(73\) | $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
73.3.2.2 | $x^{3} + 292$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.6.0.1 | $x^{6} + 45 x^{3} + 23 x^{2} + 48 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(229\) | Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
Deg $6$ | $2$ | $3$ | $3$ |