Normalized defining polynomial
\( x^{15} + 6 x^{13} - 4 x^{12} + 9 x^{11} + 3 x^{10} - 18 x^{9} + 45 x^{8} - 126 x^{7} + 216 x^{6} - 324 x^{5} + 243 x^{4} - 205 x^{3} + 90 x^{2} - 30 x + 11 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(7269488039573383941\) \(\medspace = 3^{15}\cdot 47^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.09\) | 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: | $3^{241/162}47^{1/2}\approx 35.143128879205776$ | ||
Ramified primes: | \(3\), \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{141}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{25\!\cdots\!31}a^{14}-\frac{258750044004950}{25\!\cdots\!31}a^{13}+\frac{110910697476049}{25\!\cdots\!31}a^{12}-\frac{10\!\cdots\!27}{25\!\cdots\!31}a^{11}-\frac{533127928511868}{25\!\cdots\!31}a^{10}-\frac{234487009736970}{25\!\cdots\!31}a^{9}-\frac{433602086469113}{25\!\cdots\!31}a^{8}-\frac{47648693997309}{25\!\cdots\!31}a^{7}-\frac{706262700091971}{25\!\cdots\!31}a^{6}+\frac{185251441205242}{25\!\cdots\!31}a^{5}+\frac{10\!\cdots\!17}{25\!\cdots\!31}a^{4}+\frac{164469805584897}{25\!\cdots\!31}a^{3}-\frac{460854777520338}{25\!\cdots\!31}a^{2}-\frac{734868361836927}{25\!\cdots\!31}a-\frac{11\!\cdots\!74}{25\!\cdots\!31}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{38043001556496}{25\!\cdots\!31}a^{14}-\frac{48296423944521}{25\!\cdots\!31}a^{13}+\frac{174067564617403}{25\!\cdots\!31}a^{12}-\frac{453065650852761}{25\!\cdots\!31}a^{11}+\frac{228271763832279}{25\!\cdots\!31}a^{10}-\frac{163788039386616}{25\!\cdots\!31}a^{9}-\frac{11\!\cdots\!55}{25\!\cdots\!31}a^{8}+\frac{22\!\cdots\!64}{25\!\cdots\!31}a^{7}-\frac{59\!\cdots\!13}{25\!\cdots\!31}a^{6}+\frac{12\!\cdots\!47}{25\!\cdots\!31}a^{5}-\frac{16\!\cdots\!48}{25\!\cdots\!31}a^{4}+\frac{15\!\cdots\!02}{25\!\cdots\!31}a^{3}-\frac{52\!\cdots\!32}{25\!\cdots\!31}a^{2}+\frac{67\!\cdots\!10}{25\!\cdots\!31}a-\frac{10\!\cdots\!09}{25\!\cdots\!31}$, $\frac{165444378283260}{25\!\cdots\!31}a^{14}+\frac{118218925645920}{25\!\cdots\!31}a^{13}+\frac{994029282971125}{25\!\cdots\!31}a^{12}+\frac{25592013915471}{25\!\cdots\!31}a^{11}+\frac{10\!\cdots\!04}{25\!\cdots\!31}a^{10}+\frac{14\!\cdots\!88}{25\!\cdots\!31}a^{9}-\frac{24\!\cdots\!98}{25\!\cdots\!31}a^{8}+\frac{51\!\cdots\!50}{25\!\cdots\!31}a^{7}-\frac{15\!\cdots\!34}{25\!\cdots\!31}a^{6}+\frac{21\!\cdots\!57}{25\!\cdots\!31}a^{5}-\frac{29\!\cdots\!29}{25\!\cdots\!31}a^{4}+\frac{46\!\cdots\!43}{25\!\cdots\!31}a^{3}-\frac{11\!\cdots\!26}{25\!\cdots\!31}a^{2}-\frac{19\!\cdots\!32}{25\!\cdots\!31}a+\frac{517043990670607}{25\!\cdots\!31}$, $\frac{74039452196061}{25\!\cdots\!31}a^{14}+\frac{18940358034794}{25\!\cdots\!31}a^{13}+\frac{360630633517514}{25\!\cdots\!31}a^{12}-\frac{221792748300561}{25\!\cdots\!31}a^{11}+\frac{118616125900557}{25\!\cdots\!31}a^{10}+\frac{513002582316782}{25\!\cdots\!31}a^{9}-\frac{16\!\cdots\!54}{25\!\cdots\!31}a^{8}+\frac{23\!\cdots\!73}{25\!\cdots\!31}a^{7}-\frac{69\!\cdots\!57}{25\!\cdots\!31}a^{6}+\frac{10\!\cdots\!90}{25\!\cdots\!31}a^{5}-\frac{11\!\cdots\!77}{25\!\cdots\!31}a^{4}-\frac{355338767148790}{25\!\cdots\!31}a^{3}+\frac{30\!\cdots\!31}{25\!\cdots\!31}a^{2}+\frac{15\!\cdots\!31}{25\!\cdots\!31}a+\frac{556871188999820}{25\!\cdots\!31}$, $\frac{167490929200191}{25\!\cdots\!31}a^{14}+\frac{2685719722084}{25\!\cdots\!31}a^{13}+\frac{981533778688417}{25\!\cdots\!31}a^{12}-\frac{658746539489490}{25\!\cdots\!31}a^{11}+\frac{13\!\cdots\!05}{25\!\cdots\!31}a^{10}+\frac{597140581676474}{25\!\cdots\!31}a^{9}-\frac{31\!\cdots\!54}{25\!\cdots\!31}a^{8}+\frac{74\!\cdots\!05}{25\!\cdots\!31}a^{7}-\frac{20\!\cdots\!03}{25\!\cdots\!31}a^{6}+\frac{35\!\cdots\!61}{25\!\cdots\!31}a^{5}-\frac{51\!\cdots\!48}{25\!\cdots\!31}a^{4}+\frac{35\!\cdots\!37}{25\!\cdots\!31}a^{3}-\frac{27\!\cdots\!52}{25\!\cdots\!31}a^{2}+\frac{98\!\cdots\!57}{25\!\cdots\!31}a-\frac{20\!\cdots\!31}{25\!\cdots\!31}$, $\frac{68851530394218}{25\!\cdots\!31}a^{14}+\frac{9838036202774}{25\!\cdots\!31}a^{13}+\frac{392241993717571}{25\!\cdots\!31}a^{12}-\frac{202638486436899}{25\!\cdots\!31}a^{11}+\frac{474757950087996}{25\!\cdots\!31}a^{10}+\frac{469812445896634}{25\!\cdots\!31}a^{9}-\frac{13\!\cdots\!10}{25\!\cdots\!31}a^{8}+\frac{29\!\cdots\!16}{25\!\cdots\!31}a^{7}-\frac{76\!\cdots\!12}{25\!\cdots\!31}a^{6}+\frac{12\!\cdots\!69}{25\!\cdots\!31}a^{5}-\frac{17\!\cdots\!81}{25\!\cdots\!31}a^{4}+\frac{72\!\cdots\!08}{25\!\cdots\!31}a^{3}-\frac{48\!\cdots\!86}{25\!\cdots\!31}a^{2}-\frac{17\!\cdots\!90}{25\!\cdots\!31}a-\frac{319902043572305}{25\!\cdots\!31}$, $\frac{55065280286527}{25\!\cdots\!31}a^{14}+\frac{17721662610740}{25\!\cdots\!31}a^{13}+\frac{368960415844164}{25\!\cdots\!31}a^{12}-\frac{55407559622651}{25\!\cdots\!31}a^{11}+\frac{675663729475804}{25\!\cdots\!31}a^{10}+\frac{428482794515119}{25\!\cdots\!31}a^{9}-\frac{817335203135992}{25\!\cdots\!31}a^{8}+\frac{22\!\cdots\!80}{25\!\cdots\!31}a^{7}-\frac{66\!\cdots\!21}{25\!\cdots\!31}a^{6}+\frac{10\!\cdots\!16}{25\!\cdots\!31}a^{5}-\frac{17\!\cdots\!08}{25\!\cdots\!31}a^{4}+\frac{10\!\cdots\!36}{25\!\cdots\!31}a^{3}-\frac{11\!\cdots\!55}{25\!\cdots\!31}a^{2}+\frac{17\!\cdots\!16}{25\!\cdots\!31}a-\frac{11\!\cdots\!16}{25\!\cdots\!31}$, $\frac{118218925645920}{25\!\cdots\!31}a^{14}+\frac{1363013271565}{25\!\cdots\!31}a^{13}+\frac{687369527048511}{25\!\cdots\!31}a^{12}-\frac{458407366349136}{25\!\cdots\!31}a^{11}+\frac{941386107916708}{25\!\cdots\!31}a^{10}+\frac{497934527851782}{25\!\cdots\!31}a^{9}-\frac{22\!\cdots\!50}{25\!\cdots\!31}a^{8}+\frac{52\!\cdots\!26}{25\!\cdots\!31}a^{7}-\frac{14\!\cdots\!03}{25\!\cdots\!31}a^{6}+\frac{23\!\cdots\!11}{25\!\cdots\!31}a^{5}-\frac{35\!\cdots\!37}{25\!\cdots\!31}a^{4}+\frac{22\!\cdots\!74}{25\!\cdots\!31}a^{3}-\frac{16\!\cdots\!32}{25\!\cdots\!31}a^{2}+\frac{54\!\cdots\!07}{25\!\cdots\!31}a+\frac{689442018940871}{25\!\cdots\!31}$, $\frac{110597019265331}{25\!\cdots\!31}a^{14}+\frac{76870185919336}{25\!\cdots\!31}a^{13}+\frac{710278662309861}{25\!\cdots\!31}a^{12}+\frac{53215248499520}{25\!\cdots\!31}a^{11}+\frac{965539067417517}{25\!\cdots\!31}a^{10}+\frac{10\!\cdots\!03}{25\!\cdots\!31}a^{9}-\frac{14\!\cdots\!36}{25\!\cdots\!31}a^{8}+\frac{41\!\cdots\!59}{25\!\cdots\!31}a^{7}-\frac{10\!\cdots\!69}{25\!\cdots\!31}a^{6}+\frac{15\!\cdots\!49}{25\!\cdots\!31}a^{5}-\frac{23\!\cdots\!73}{25\!\cdots\!31}a^{4}+\frac{75\!\cdots\!21}{25\!\cdots\!31}a^{3}-\frac{10\!\cdots\!80}{25\!\cdots\!31}a^{2}-\frac{18\!\cdots\!60}{25\!\cdots\!31}a-\frac{19\!\cdots\!85}{25\!\cdots\!31}$ | 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: | \( 2652.22907153 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 2652.22907153 \cdot 1}{2\cdot\sqrt{7269488039573383941}}\cr\approx \mathstrut & 0.242101967819 \end{aligned}\]
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
5.1.2209.1 |
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: | 15.1.154669958288795403.3 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | 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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.15.15.38 | $x^{15} + 30 x^{14} + 432 x^{13} + 4335 x^{12} + 32373 x^{11} + 162027 x^{10} + 501129 x^{9} + 945432 x^{8} + 1104921 x^{7} + 951939 x^{6} + 550233 x^{5} + 242271 x^{4} + 60669 x^{3} + 11664 x^{2} - 729 x + 243$ | $3$ | $5$ | $15$ | 15T33 | $[3/2, 3/2, 3/2, 3/2]_{2}^{5}$ |
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
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 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |