Normalized defining polynomial
\( x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + \cdots - 1 \)
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)
| |
Discriminant: | \(-35351257235385344\) \(\medspace = -\,2^{10}\cdot 11^{13}\) | 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: | \(12.68\) | 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: | $2^{2/3}11^{9/10}\approx 13.738524116073206$ | ||
Ramified primes: | \(2\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2769251}a^{14}+\frac{1237085}{2769251}a^{13}-\frac{190661}{2769251}a^{12}-\frac{1217018}{2769251}a^{11}+\frac{1209397}{2769251}a^{10}+\frac{265482}{2769251}a^{9}-\frac{1060863}{2769251}a^{8}+\frac{806293}{2769251}a^{7}-\frac{772551}{2769251}a^{6}-\frac{1194966}{2769251}a^{5}-\frac{1253042}{2769251}a^{4}-\frac{753537}{2769251}a^{3}-\frac{1278776}{2769251}a^{2}-\frac{589461}{2769251}a-\frac{341613}{2769251}$
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)
| |
Fundamental units: | $\frac{516353}{2769251}a^{14}+\frac{499839}{2769251}a^{13}-\frac{1506283}{2769251}a^{12}-\frac{4150681}{2769251}a^{11}+\frac{1360888}{2769251}a^{10}+\frac{4502646}{2769251}a^{9}+\frac{8516922}{2769251}a^{8}-\frac{155162}{2769251}a^{7}-\frac{15035459}{2769251}a^{6}-\frac{4694437}{2769251}a^{5}+\frac{1346316}{2769251}a^{4}+\frac{10828947}{2769251}a^{3}+\frac{8692265}{2769251}a^{2}-\frac{7116825}{2769251}a-\frac{2685693}{2769251}$, $\frac{975382}{2769251}a^{14}+\frac{1318746}{2769251}a^{13}-\frac{1025848}{2769251}a^{12}-\frac{6932722}{2769251}a^{11}-\frac{2091569}{2769251}a^{10}-\frac{758384}{2769251}a^{9}+\frac{17192496}{2769251}a^{8}+\frac{6856687}{2769251}a^{7}-\frac{5296127}{2769251}a^{6}-\frac{14889128}{2769251}a^{5}-\frac{16144955}{2769251}a^{4}+\frac{6020278}{2769251}a^{3}+\frac{8788980}{2769251}a^{2}+\frac{10551271}{2769251}a+\frac{1416907}{2769251}$, $\frac{821903}{2769251}a^{14}+\frac{137093}{2769251}a^{13}-\frac{1241546}{2769251}a^{12}-\frac{3437799}{2769251}a^{11}+\frac{3760798}{2769251}a^{10}-\frac{5449550}{2769251}a^{9}+\frac{13733826}{2769251}a^{8}-\frac{9282729}{2769251}a^{7}+\frac{2346488}{2769251}a^{6}+\frac{1957864}{2769251}a^{5}-\frac{13916783}{2769251}a^{4}+\frac{11434490}{2769251}a^{3}-\frac{10460196}{2769251}a^{2}+\frac{9005920}{2769251}a+\frac{4378602}{2769251}$, $\frac{450856}{2769251}a^{14}+\frac{658603}{2769251}a^{13}-\frac{335525}{2769251}a^{12}-\frac{3243519}{2769251}a^{11}-\frac{1628068}{2769251}a^{10}-\frac{1183381}{2769251}a^{9}+\frac{8583992}{2769251}a^{8}+\frac{5227289}{2769251}a^{7}-\frac{1170629}{2769251}a^{6}-\frac{10885850}{2769251}a^{5}-\frac{8761450}{2769251}a^{4}+\frac{3342761}{2769251}a^{3}+\frac{8687442}{2769251}a^{2}+\frac{5759105}{2769251}a-\frac{837861}{2769251}$, $\frac{137093}{2769251}a^{14}+\frac{1224163}{2769251}a^{13}+\frac{671716}{2769251}a^{12}-\frac{2814426}{2769251}a^{11}-\frac{6271453}{2769251}a^{10}+\frac{2227184}{2769251}a^{9}+\frac{1402010}{2769251}a^{8}+\frac{16318839}{2769251}a^{7}-\frac{1329748}{2769251}a^{6}-\frac{14738686}{2769251}a^{5}-\frac{6647376}{2769251}a^{4}-\frac{8816390}{2769251}a^{3}+\frac{15581144}{2769251}a^{2}+\frac{6844311}{2769251}a+\frac{821903}{2769251}$, $\frac{30196}{2769251}a^{14}+\frac{591921}{2769251}a^{13}+\frac{73273}{2769251}a^{12}-\frac{1114758}{2769251}a^{11}-\frac{1930376}{2769251}a^{10}+\frac{2282078}{2769251}a^{9}-\frac{4662082}{2769251}a^{8}+\frac{7876389}{2769251}a^{7}+\frac{220428}{2769251}a^{6}-\frac{2622057}{2769251}a^{5}+\frac{2189432}{2769251}a^{4}-\frac{9944789}{2769251}a^{3}+\frac{515848}{2769251}a^{2}+\frac{4150323}{2769251}a+\frac{2883078}{2769251}$, $\frac{108007}{2769251}a^{14}+\frac{248096}{2769251}a^{13}-\frac{572191}{2769251}a^{12}-\frac{1195160}{2769251}a^{11}+\frac{541360}{2769251}a^{10}+\frac{3858771}{2769251}a^{9}-\frac{100665}{2769251}a^{8}+\frac{651854}{2769251}a^{7}-\frac{11690980}{2769251}a^{6}+\frac{1788595}{2769251}a^{5}+\frac{7066080}{2769251}a^{4}+\frac{3785382}{2769251}a^{3}+\frac{5172695}{2769251}a^{2}-\frac{11910741}{2769251}a-\frac{1864218}{2769251}$, $\frac{37727}{2769251}a^{14}-\frac{1450559}{2769251}a^{13}-\frac{1322700}{2769251}a^{12}+\frac{2512745}{2769251}a^{11}+\frac{9048896}{2769251}a^{10}-\frac{3310704}{2769251}a^{9}+\frac{806302}{2769251}a^{8}-\frac{20590981}{2769251}a^{7}+\frac{335198}{2769251}a^{6}+\frac{14770253}{2769251}a^{5}+\frac{5906789}{2769251}a^{4}+\frac{11517371}{2769251}a^{3}-\frac{15106736}{2769251}a^{2}-\frac{7048119}{2769251}a+\frac{60503}{2769251}$, $\frac{1659764}{2769251}a^{14}-\frac{313763}{2769251}a^{13}-\frac{4413732}{2769251}a^{12}-\frac{7291579}{2769251}a^{11}+\frac{13707205}{2769251}a^{10}-\frac{2983621}{2769251}a^{9}+\frac{26647261}{2769251}a^{8}-\frac{28758914}{2769251}a^{7}-\frac{17124438}{2769251}a^{6}+\frac{6479937}{2769251}a^{5}-\frac{2193072}{2769251}a^{4}+\frac{38628882}{2769251}a^{3}-\frac{15478679}{2769251}a^{2}-\frac{6384410}{2769251}a-\frac{3894086}{2769251}$ | 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: | \( 211.33204879 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 211.33204879 \cdot 1}{2\cdot\sqrt{35351257235385344}}\cr\approx \mathstrut & 0.17610928587 \end{aligned}\]
Galois group
$C_5\times S_3$ (as 15T4):
A solvable group of order 30 |
The 15 conjugacy class representatives for $S_3 \times C_5$ |
Character table for $S_3 \times C_5$ |
Intermediate fields
3.1.44.1, \(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 30 |
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 | $15$ | $15$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.3.0.1}{3} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | $15$ | $15$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | $15$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.15.10.1 | $x^{15} + 13 x^{12} + 3 x^{10} + 13 x^{9} - 174 x^{7} - 45 x^{6} + 3 x^{5} + 513 x^{4} + 63 x^{2} - 216 x + 109$ | $3$ | $5$ | $10$ | $S_3 \times C_5$ | $[\ ]_{3}^{10}$ |
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
1.11.10t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
1.11.10t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
1.11.10t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
* | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
1.11.10t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
* | 2.44.3t2.b.a | $2$ | $ 2^{2} \cdot 11 $ | 3.1.44.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.484.15t4.a.a | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.35351257235385344.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
* | 2.484.15t4.a.b | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.35351257235385344.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
* | 2.484.15t4.a.c | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.35351257235385344.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
* | 2.484.15t4.a.d | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.35351257235385344.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |