Normalized defining polynomial
\( x^{15} - 15 x^{13} - x^{12} + 79 x^{11} + 25 x^{10} - 219 x^{9} - 173 x^{8} + 611 x^{7} + 265 x^{6} + \cdots + 4 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[9, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-349046216993499621221376\) \(\medspace = -\,2^{10}\cdot 3^{6}\cdot 881^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.11\) | 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: | $2^{5/3}3^{7/6}881^{2/3}\approx 1051.1759004017842$ | ||
Ramified primes: | \(2\), \(3\), \(881\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{6}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{311705569432}a^{14}+\frac{4093683000}{38963196179}a^{13}-\frac{31682909139}{311705569432}a^{12}-\frac{71812695063}{311705569432}a^{11}-\frac{67584744459}{311705569432}a^{10}-\frac{77728241749}{311705569432}a^{9}+\frac{67931698931}{311705569432}a^{8}+\frac{25214003789}{311705569432}a^{7}-\frac{145227726055}{311705569432}a^{6}+\frac{20225838803}{311705569432}a^{5}-\frac{35716485693}{311705569432}a^{4}+\frac{51394379049}{311705569432}a^{3}-\frac{13371921365}{77926392358}a^{2}+\frac{99215260847}{311705569432}a-\frac{7234544315}{155852784716}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | 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: | $\frac{13870226639}{311705569432}a^{14}+\frac{1601116588}{38963196179}a^{13}-\frac{189111396523}{311705569432}a^{12}-\frac{186192257891}{311705569432}a^{11}+\frac{830746823541}{311705569432}a^{10}+\frac{1065889043647}{311705569432}a^{9}-\frac{1640796360577}{311705569432}a^{8}-\frac{3543508349939}{311705569432}a^{7}+\frac{4223957982561}{311705569432}a^{6}+\frac{6188132853031}{311705569432}a^{5}-\frac{9411336594301}{311705569432}a^{4}+\frac{800983852565}{311705569432}a^{3}+\frac{965025956137}{155852784716}a^{2}-\frac{1842115956313}{311705569432}a+\frac{310345755997}{155852784716}$, $\frac{2473927943}{77926392358}a^{14}+\frac{422044209}{77926392358}a^{13}-\frac{76198203509}{155852784716}a^{12}-\frac{20594230725}{155852784716}a^{11}+\frac{417216002971}{155852784716}a^{10}+\frac{232702729725}{155852784716}a^{9}-\frac{1177293543087}{155852784716}a^{8}-\frac{1265638568059}{155852784716}a^{7}+\frac{3061874015383}{155852784716}a^{6}+\frac{2514643121149}{155852784716}a^{5}-\frac{6532360136111}{155852784716}a^{4}+\frac{121672872459}{155852784716}a^{3}+\frac{3626648442077}{155852784716}a^{2}-\frac{198495753919}{155852784716}a-\frac{147369518761}{77926392358}$, $\frac{2312494921}{155852784716}a^{14}+\frac{754123355}{155852784716}a^{13}-\frac{18616841651}{77926392358}a^{12}-\frac{4137194641}{38963196179}a^{11}+\frac{106551302817}{77926392358}a^{10}+\frac{41734095083}{38963196179}a^{9}-\frac{148959967036}{38963196179}a^{8}-\frac{431302920625}{77926392358}a^{7}+\frac{338894375972}{38963196179}a^{6}+\frac{468819876990}{38963196179}a^{5}-\frac{1347635617291}{77926392358}a^{4}-\frac{210032201420}{38963196179}a^{3}+\frac{1316973733071}{155852784716}a^{2}+\frac{327783123595}{155852784716}a-\frac{88813987}{77926392358}$, $\frac{10767443479}{311705569432}a^{14}+\frac{4423979329}{155852784716}a^{13}-\frac{153842638579}{311705569432}a^{12}-\frac{138132086539}{311705569432}a^{11}+\frac{731840147101}{311705569432}a^{10}+\frac{888718782547}{311705569432}a^{9}-\frac{1601384593401}{311705569432}a^{8}-\frac{3283468132515}{311705569432}a^{7}+\frac{3762593851645}{311705569432}a^{6}+\frac{6200636052215}{311705569432}a^{5}-\frac{8107399897957}{311705569432}a^{4}-\frac{1639500087239}{311705569432}a^{3}+\frac{1129134238825}{155852784716}a^{2}+\frac{332975912197}{311705569432}a+\frac{428227060267}{155852784716}$, $\frac{17713303799}{311705569432}a^{14}-\frac{511970031}{38963196179}a^{13}-\frac{273367233909}{311705569432}a^{12}+\frac{38941350463}{311705569432}a^{11}+\frac{1494150223691}{311705569432}a^{10}+\frac{205011057881}{311705569432}a^{9}-\frac{4307311162375}{311705569432}a^{8}-\frac{2691887883873}{311705569432}a^{7}+\frac{12089581345367}{311705569432}a^{6}+\frac{3645188311557}{311705569432}a^{5}-\frac{25339887979851}{311705569432}a^{4}+\frac{12520825906259}{311705569432}a^{3}+\frac{967147439529}{38963196179}a^{2}-\frac{6294744117059}{311705569432}a+\frac{626292689343}{155852784716}$, $\frac{3230413}{1002268712}a^{14}+\frac{5069313}{501134356}a^{13}-\frac{27517023}{1002268712}a^{12}-\frac{141354487}{1002268712}a^{11}-\frac{35951711}{1002268712}a^{10}+\frac{620501487}{1002268712}a^{9}+\frac{743543475}{1002268712}a^{8}-\frac{918830587}{1002268712}a^{7}-\frac{1861083839}{1002268712}a^{6}+\frac{963810515}{1002268712}a^{5}+\frac{2512159887}{1002268712}a^{4}-\frac{3263942019}{1002268712}a^{3}-\frac{78767634}{125283589}a^{2}+\frac{1202957765}{1002268712}a-\frac{188421793}{501134356}$, $\frac{17162157679}{311705569432}a^{14}+\frac{10225210725}{77926392358}a^{13}-\frac{203523167559}{311705569432}a^{12}-\frac{571068742407}{311705569432}a^{11}+\frac{573713385037}{311705569432}a^{10}+\frac{2779826137715}{311705569432}a^{9}+\frac{537476433443}{311705569432}a^{8}-\frac{6878243888387}{311705569432}a^{7}-\frac{2795464390339}{311705569432}a^{6}+\frac{13052170222187}{311705569432}a^{5}+\frac{3430733701515}{311705569432}a^{4}-\frac{12095322850847}{311705569432}a^{3}-\frac{2285880003613}{155852784716}a^{2}+\frac{1460480740179}{311705569432}a+\frac{65796736721}{155852784716}$, $\frac{8403779513}{311705569432}a^{14}+\frac{1605339761}{155852784716}a^{13}-\frac{125411237311}{311705569432}a^{12}-\frac{61264417823}{311705569432}a^{11}+\frac{647849958693}{311705569432}a^{10}+\frac{508300207011}{311705569432}a^{9}-\frac{1662572823953}{311705569432}a^{8}-\frac{2250292233291}{311705569432}a^{7}+\frac{4182387388937}{311705569432}a^{6}+\frac{4184886052115}{311705569432}a^{5}-\frac{8951687582229}{311705569432}a^{4}+\frac{91127623137}{311705569432}a^{3}+\frac{543862156987}{38963196179}a^{2}-\frac{831097577291}{311705569432}a-\frac{44873280597}{155852784716}$, $\frac{6838498885}{311705569432}a^{14}+\frac{4280659067}{155852784716}a^{13}-\frac{90248556113}{311705569432}a^{12}-\frac{121798846669}{311705569432}a^{11}+\frac{361324030723}{311705569432}a^{10}+\frac{656573530789}{311705569432}a^{9}-\frac{525742430971}{311705569432}a^{8}-\frac{1993483823265}{311705569432}a^{7}+\frac{1211168223351}{311705569432}a^{6}+\frac{3474900268721}{311705569432}a^{5}-\frac{2819261730107}{311705569432}a^{4}-\frac{501758982393}{311705569432}a^{3}-\frac{273108151917}{155852784716}a^{2}+\frac{835443062143}{311705569432}a-\frac{109376800251}{155852784716}$, $\frac{7965075039}{155852784716}a^{14}+\frac{2562953365}{155852784716}a^{13}-\frac{57440199349}{77926392358}a^{12}-\frac{9476920788}{38963196179}a^{11}+\frac{285230182019}{77926392358}a^{10}+\frac{147481395311}{77926392358}a^{9}-\frac{370315089397}{38963196179}a^{8}-\frac{364495362574}{38963196179}a^{7}+\frac{1049687106940}{38963196179}a^{6}+\frac{587290454054}{38963196179}a^{5}-\frac{4468278270307}{77926392358}a^{4}+\frac{1771697067533}{77926392358}a^{3}+\frac{3072040157021}{155852784716}a^{2}-\frac{2011702969609}{155852784716}a-\frac{27917734879}{77926392358}$, $\frac{24855013}{155852784716}a^{14}+\frac{853149231}{155852784716}a^{13}+\frac{37674362}{38963196179}a^{12}-\frac{2689413049}{38963196179}a^{11}-\frac{2428209663}{38963196179}a^{10}+\frac{24501215569}{77926392358}a^{9}+\frac{19298421448}{38963196179}a^{8}-\frac{68242581947}{77926392358}a^{7}-\frac{110777683615}{77926392358}a^{6}+\frac{78935398359}{38963196179}a^{5}+\frac{69451011765}{38963196179}a^{4}-\frac{155224551913}{77926392358}a^{3}-\frac{200781972805}{155852784716}a^{2}+\frac{170850409695}{155852784716}a-\frac{10190670243}{77926392358}$ | 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: | \( 9545897.66025 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{3}\cdot 9545897.66025 \cdot 1}{2\cdot\sqrt{349046216993499621221376}}\cr\approx \mathstrut & 1.02601775626 \end{aligned}\]
Galois group
$S_3\wr A_5$ (as 15T90):
A non-solvable group of order 466560 |
The 72 conjugacy class representatives for $S_3\wr A_5$ |
Character table for $S_3\wr A_5$ |
Intermediate fields
5.5.3104644.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 30 siblings: | data not computed |
Degree 45 sibling: | 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 | R | R | $15$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | $15$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | $15$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(881\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |