Normalized defining polynomial
\( x^{15} - 20x^{9} - 27x^{6} - 10x^{3} - 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-59660248299440788525707\) \(\medspace = -\,3^{15}\cdot 401^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.99\) | 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: | not computed | ||
Ramified primes: | \(3\), \(401\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | 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: | $a$, $2a^{13}-\frac{2}{3}a^{10}-\frac{119}{3}a^{7}-41a^{4}-\frac{23}{3}a$, $a^{14}-a^{11}-19a^{8}-8a^{5}-2a^{2}$, $a^{14}+a^{13}-a^{11}-a^{10}-19a^{8}-19a^{7}-8a^{5}-8a^{4}-2a^{2}-2a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{9}-7a^{6}-\frac{47}{3}a^{3}-5$, $a^{14}-\frac{8}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+2a^{10}+\frac{2}{3}a^{9}-\frac{62}{3}a^{8}+\frac{155}{3}a^{7}+6a^{6}-33a^{5}+\frac{100}{3}a^{4}-\frac{10}{3}a^{3}-\frac{20}{3}a^{2}+\frac{14}{3}a-1$, $\frac{13}{3}a^{14}-\frac{7}{3}a^{13}+a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{260}{3}a^{8}+\frac{140}{3}a^{7}-20a^{6}-\frac{332}{3}a^{5}+\frac{167}{3}a^{4}-27a^{3}-\frac{98}{3}a^{2}+\frac{50}{3}a-9$, $\frac{44}{3}a^{14}+\frac{47}{3}a^{13}+\frac{1}{3}a^{12}-\frac{5}{3}a^{11}-3a^{10}+\frac{1}{3}a^{9}-\frac{880}{3}a^{8}-\frac{938}{3}a^{7}-7a^{6}-\frac{1087}{3}a^{5}-\frac{1090}{3}a^{4}-\frac{47}{3}a^{3}-\frac{304}{3}a^{2}-\frac{266}{3}a-8$, $\frac{14}{3}a^{14}+\frac{19}{3}a^{13}-2a^{11}-\frac{7}{3}a^{10}-\frac{278}{3}a^{8}-\frac{377}{3}a^{7}-\frac{259}{3}a^{5}-\frac{374}{3}a^{4}-\frac{20}{3}a^{2}-\frac{56}{3}a+1$ (assuming GRH) | 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: | \( 568023.433984 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 568023.433984 \cdot 1}{2\cdot\sqrt{59660248299440788525707}}\cr\approx \mathstrut & 0.364370366309 \end{aligned}\] (assuming GRH)
Galois group
$C_3^4:D_{10}$ (as 15T43):
A solvable group of order 1620 |
The 24 conjugacy class representatives for $C_3^4:D_{10}$ |
Character table for $C_3^4:D_{10}$ is not computed |
Intermediate fields
5.5.160801.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 15 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 45 siblings: | data not computed |
Minimal sibling: | 15.5.6628916477715643169523.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.5.0.1}{5} }^{3}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
3.6.6.4 | $x^{6} + 48 x^{4} + 6 x^{3} + 36 x^{2} + 36 x + 9$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(401\) | $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |