Normalized defining polynomial
\( x^{15} - 10x^{12} + 29x^{9} - 25x^{6} + 3x^{3} + 1 \)
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))
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23}a^{12}+\frac{11}{23}a^{9}+\frac{7}{23}a^{6}+\frac{7}{23}a^{3}-\frac{11}{23}$, $\frac{1}{23}a^{13}+\frac{11}{23}a^{10}+\frac{7}{23}a^{7}+\frac{7}{23}a^{4}-\frac{11}{23}a$, $\frac{1}{23}a^{14}+\frac{11}{23}a^{11}+\frac{7}{23}a^{8}+\frac{7}{23}a^{5}-\frac{11}{23}a^{2}$
Monogenic: | Not computed | |
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: | $a$, $\frac{1}{23}a^{14}-\frac{12}{23}a^{11}+\frac{53}{23}a^{8}-\frac{108}{23}a^{5}+\frac{81}{23}a^{2}$, $\frac{4}{23}a^{14}-\frac{48}{23}a^{11}+\frac{189}{23}a^{8}-\frac{271}{23}a^{5}+\frac{94}{23}a^{2}$, $\frac{1}{23}a^{13}-\frac{12}{23}a^{10}+\frac{53}{23}a^{7}-\frac{108}{23}a^{4}+\frac{58}{23}a$, $a^{14}+\frac{1}{23}a^{13}-10a^{11}-\frac{12}{23}a^{10}+29a^{8}+\frac{53}{23}a^{7}-25a^{5}-\frac{108}{23}a^{4}+3a^{2}+\frac{81}{23}a$, $\frac{3}{23}a^{14}+\frac{3}{23}a^{13}+\frac{3}{23}a^{12}-\frac{36}{23}a^{11}-\frac{36}{23}a^{10}-\frac{36}{23}a^{9}+\frac{136}{23}a^{8}+\frac{136}{23}a^{7}+\frac{136}{23}a^{6}-\frac{163}{23}a^{5}-\frac{163}{23}a^{4}-\frac{163}{23}a^{3}+\frac{13}{23}a^{2}+\frac{13}{23}a+\frac{13}{23}$, $\frac{9}{23}a^{13}+\frac{2}{23}a^{12}-\frac{85}{23}a^{10}-\frac{24}{23}a^{9}+\frac{224}{23}a^{7}+\frac{83}{23}a^{6}-\frac{167}{23}a^{4}-\frac{78}{23}a^{3}+\frac{16}{23}a+\frac{24}{23}$, $\frac{5}{23}a^{14}-\frac{11}{23}a^{13}+\frac{1}{23}a^{12}-\frac{60}{23}a^{11}+\frac{109}{23}a^{10}-\frac{12}{23}a^{9}+\frac{242}{23}a^{8}-\frac{307}{23}a^{7}+\frac{53}{23}a^{6}-\frac{379}{23}a^{5}+\frac{245}{23}a^{4}-\frac{108}{23}a^{3}+\frac{152}{23}a^{2}-\frac{63}{23}a+\frac{35}{23}$, $\frac{1}{23}a^{14}+\frac{8}{23}a^{12}-\frac{12}{23}a^{11}-\frac{73}{23}a^{9}+\frac{53}{23}a^{8}+\frac{171}{23}a^{6}-\frac{108}{23}a^{5}-\frac{59}{23}a^{3}+\frac{81}{23}a^{2}+a-\frac{42}{23}$ | 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: | \( 8686.79947771 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 8686.79947771 \cdot 1}{2\cdot\sqrt{45033016903730593947}}\cr\approx \mathstrut & 0.202821317573 \end{aligned}\]
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: | 15.5.45033016903730593947.3 |
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 | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.15.15.28 | $x^{15} + 381 x^{13} + 1383 x^{12} + 11340 x^{11} + 18261 x^{10} - 15192 x^{9} + 2916 x^{8} + 216837 x^{7} + 473310 x^{6} + 423549 x^{5} + 128871 x^{4} + 23733 x^{3} + 10206 x^{2} - 2673 x + 243$ | $3$ | $5$ | $15$ | 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 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |