Normalized defining polynomial
\( x^{15} - 5x^{12} + 10x^{9} - 13x^{6} + 13x^{3} - 5 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-3866748957219885075\) \(\medspace = -\,3^{15}\cdot 5^{2}\cdot 47^{6}\) | 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: | \(17.34\) | 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: | $3^{727/486}5^{2/3}47^{1/2}\approx 103.69250158786609$ | ||
Ramified primes: | \(3\), \(5\), \(47\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{9}-\frac{1}{5}a^{6}-\frac{1}{5}a^{3}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{10}-\frac{1}{5}a^{7}-\frac{1}{5}a^{4}$, $\frac{1}{5}a^{14}-\frac{2}{5}a^{11}-\frac{1}{5}a^{8}-\frac{1}{5}a^{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{2}{5}a^{12}-\frac{9}{5}a^{9}+\frac{13}{5}a^{6}-\frac{12}{5}a^{3}+2$, $\frac{1}{5}a^{12}-\frac{7}{5}a^{9}+\frac{14}{5}a^{6}-\frac{16}{5}a^{3}+3$, $\frac{2}{5}a^{14}+\frac{3}{5}a^{13}-\frac{1}{5}a^{12}-\frac{9}{5}a^{11}-\frac{11}{5}a^{10}+\frac{2}{5}a^{9}+\frac{13}{5}a^{8}+\frac{17}{5}a^{7}+\frac{1}{5}a^{6}-\frac{12}{5}a^{5}-\frac{18}{5}a^{4}+\frac{1}{5}a^{3}+2a^{2}+2a$, $\frac{2}{5}a^{14}-\frac{1}{5}a^{13}+\frac{1}{5}a^{12}-\frac{9}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{9}+\frac{13}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{12}{5}a^{5}-\frac{4}{5}a^{4}+\frac{4}{5}a^{3}+2a^{2}+a$, $\frac{4}{5}a^{13}-\frac{1}{5}a^{12}-\frac{13}{5}a^{10}+\frac{2}{5}a^{9}+\frac{16}{5}a^{7}-\frac{4}{5}a^{6}-\frac{19}{5}a^{4}+\frac{6}{5}a^{3}+a^{2}+3a$, $\frac{1}{5}a^{13}-\frac{1}{5}a^{12}-\frac{7}{5}a^{10}+\frac{7}{5}a^{9}+\frac{14}{5}a^{7}-\frac{14}{5}a^{6}-\frac{16}{5}a^{4}+\frac{16}{5}a^{3}+3a-3$, $\frac{2}{5}a^{14}+\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{9}{5}a^{11}-\frac{9}{5}a^{10}-\frac{2}{5}a^{9}+\frac{18}{5}a^{8}+\frac{13}{5}a^{7}-\frac{1}{5}a^{6}-\frac{22}{5}a^{5}-\frac{12}{5}a^{4}-\frac{1}{5}a^{3}+3a^{2}+2a+1$ | 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: | \( 1346.00590418 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 1346.00590418 \cdot 1}{2\cdot\sqrt{3866748957219885075}}\cr\approx \mathstrut & 0.264626330664 \end{aligned}\]
Galois group
$C_3^5:D_{10}$ (as 15T55):
A solvable group of order 4860 |
The 48 conjugacy class representatives for $C_3^5:D_{10}$ |
Character table for $C_3^5:D_{10}$ |
Intermediate fields
5.1.2209.1 |
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: | 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 | R | $15$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | $15$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | R | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.15.15.48 | $x^{15} - 24 x^{14} + 333 x^{13} - 2856 x^{12} + 9477 x^{11} + 27486 x^{10} - 38493 x^{9} + 24705 x^{8} + 256527 x^{7} + 115533 x^{6} - 123363 x^{5} - 63909 x^{4} + 18144 x^{3} + 8019 x^{2} - 729 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |