Normalized defining polynomial
\( x^{15} - 3x^{12} - 3x^{9} + 15x^{6} - 10x^{3} - 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-45033016903730593947\) \(\medspace = -\,3^{15}\cdot 11^{12}\) | 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: | \(20.43\) | 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: | $3^{241/162}11^{4/5}\approx 34.90644688215396$ | ||
Ramified primes: | \(3\), \(11\) | 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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $a^{14}-3a^{11}-3a^{8}+15a^{5}-10a^{2}$, $3a^{14}+5a^{11}+15a^{8}-24a^{5}+a^{2}$, $2a^{13}+3a^{10}+11a^{7}-14a^{4}-4a$, $a^{13}-2a^{10}-5a^{7}+10a^{4}-a$, $3a^{14}-4a^{12}-5a^{11}+6a^{9}-15a^{8}+21a^{6}+24a^{5}-29a^{3}-a^{2}-3$, $a^{14}+2a^{13}-2a^{11}-3a^{10}-4a^{8}-11a^{7}+10a^{5}+14a^{4}-5a^{2}+4a-1$, $a^{12}-a^{9}-5a^{6}+5a^{3}-a^{2}-a+1$, $a^{14}+2a^{13}+a^{11}-3a^{10}+6a^{8}-11a^{7}-5a^{5}+14a^{4}-4a^{2}+4a-1$, $3a^{14}-3a^{13}+5a^{12}-5a^{11}+5a^{10}-8a^{9}-15a^{8}+16a^{7}-26a^{6}+24a^{5}-24a^{4}+38a^{3}-a^{2}-4a+3$ | 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: | \( 10031.504095520013 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 10031.504095520013 \cdot 1}{2\cdot\sqrt{45033016903730593947}}\cr\approx \mathstrut & 0.234217778724206 \end{aligned}\]
Galois group
$C_3^4:C_{10}$ (as 15T33):
A solvable group of order 810 |
The 18 conjugacy class representatives for $C_3^4:C_{10}$ |
Character table for $C_3^4:C_{10}$ |
Intermediate fields
\(\Q(\zeta_{11})^+\) |
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: | 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{9}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.15.15.9 | $x^{15} - 33 x^{14} + 603 x^{13} - 2118 x^{12} + 51525 x^{11} + 162432 x^{10} + 260424 x^{9} + 402408 x^{8} + 416502 x^{7} + 248859 x^{6} + 39933 x^{5} - 10449 x^{4} + 3321 x^{3} + 10449 x^{2} - 729 x + 243$ | $3$ | $5$ | $15$ | 15T33 | $[3/2, 3/2, 3/2, 3/2]_{2}^{5}$ |
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |