Normalized defining polynomial
\( x^{15} - 3 x^{14} + 2 x^{13} - x^{12} + 5 x^{11} - 22 x^{9} + 33 x^{8} - 22 x^{7} + 22 x^{6} - 22 x^{5} + \cdots - 1 \)
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)
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Discriminant: | \(-388863829589238784\) \(\medspace = -\,2^{10}\cdot 11^{14}\) | 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: | \(14.88\) | 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: | $2\cdot 11^{14/15}\approx 18.749794052793643$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{86}a^{13}-\frac{10}{43}a^{12}+\frac{1}{43}a^{11}+\frac{7}{43}a^{10}-\frac{5}{43}a^{9}-\frac{16}{43}a^{8}+\frac{9}{86}a^{7}+\frac{5}{43}a^{6}-\frac{27}{86}a^{5}-\frac{19}{43}a^{4}-\frac{16}{43}a^{2}+\frac{23}{86}a-\frac{16}{43}$, $\frac{1}{86}a^{14}-\frac{11}{86}a^{12}+\frac{11}{86}a^{11}+\frac{6}{43}a^{10}-\frac{17}{86}a^{9}-\frac{29}{86}a^{8}-\frac{25}{86}a^{7}-\frac{21}{43}a^{6}+\frac{12}{43}a^{5}-\frac{29}{86}a^{4}+\frac{11}{86}a^{3}-\frac{15}{86}a^{2}+\frac{41}{86}a+\frac{5}{86}$
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: | $\frac{333}{43}a^{14}-\frac{834}{43}a^{13}+\frac{246}{43}a^{12}-\frac{198}{43}a^{11}+\frac{1565}{43}a^{10}+\frac{787}{43}a^{9}-\frac{6963}{43}a^{8}+\frac{7518}{43}a^{7}-\frac{3492}{43}a^{6}+\frac{5484}{43}a^{5}-\frac{4539}{43}a^{4}-\frac{5367}{43}a^{3}+\frac{10083}{43}a^{2}-\frac{4583}{43}a+\frac{575}{43}$, $\frac{615}{43}a^{14}-\frac{1563}{43}a^{13}+\frac{501}{43}a^{12}-\frac{360}{43}a^{11}+\frac{2913}{43}a^{10}+\frac{1348}{43}a^{9}-\frac{12969}{43}a^{8}+\frac{14289}{43}a^{7}-\frac{6759}{43}a^{6}+\frac{10263}{43}a^{5}-\frac{8751}{43}a^{4}-\frac{9747}{43}a^{3}+\frac{18990}{43}a^{2}-\frac{8928}{43}a+\frac{1233}{43}$, $\frac{333}{43}a^{14}-\frac{834}{43}a^{13}+\frac{246}{43}a^{12}-\frac{198}{43}a^{11}+\frac{1565}{43}a^{10}+\frac{787}{43}a^{9}-\frac{6963}{43}a^{8}+\frac{7518}{43}a^{7}-\frac{3492}{43}a^{6}+\frac{5484}{43}a^{5}-\frac{4539}{43}a^{4}-\frac{5367}{43}a^{3}+\frac{10083}{43}a^{2}-\frac{4626}{43}a+\frac{618}{43}$, $a$, $\frac{353}{86}a^{14}-\frac{1033}{86}a^{13}+\frac{609}{86}a^{12}-\frac{247}{86}a^{11}+\frac{864}{43}a^{10}+\frac{115}{86}a^{9}-\frac{7883}{86}a^{8}+\frac{5516}{43}a^{7}-\frac{3204}{43}a^{6}+\frac{6779}{86}a^{5}-\frac{7103}{86}a^{4}-\frac{4029}{86}a^{3}+\frac{13485}{86}a^{2}-\frac{4385}{43}a+\frac{1711}{86}$, $\frac{1072}{43}a^{14}-65a^{13}+\frac{2087}{86}a^{12}-\frac{1313}{86}a^{11}+\frac{10205}{86}a^{10}+\frac{4015}{86}a^{9}-\frac{45621}{86}a^{8}+\frac{52825}{86}a^{7}-\frac{13161}{43}a^{6}+\frac{18375}{43}a^{5}-\frac{32721}{86}a^{4}-\frac{32273}{86}a^{3}+\frac{68933}{86}a^{2}-\frac{35119}{86}a+\frac{2694}{43}$, $\frac{1413}{86}a^{14}-\frac{3533}{86}a^{13}+\frac{533}{43}a^{12}-\frac{897}{86}a^{11}+\frac{3312}{43}a^{10}+\frac{77}{2}a^{9}-\frac{29401}{86}a^{8}+\frac{15975}{43}a^{7}-\frac{15255}{86}a^{6}+\frac{23609}{86}a^{5}-\frac{9670}{43}a^{4}-\frac{22297}{86}a^{3}+\frac{42497}{86}a^{2}-\frac{9943}{43}a+\frac{1473}{43}$, $\frac{279}{43}a^{14}-\frac{1527}{86}a^{13}+\frac{709}{86}a^{12}-\frac{178}{43}a^{11}+\frac{1345}{43}a^{10}+\frac{355}{43}a^{9}-\frac{6062}{43}a^{8}+\frac{15221}{86}a^{7}-\frac{8133}{86}a^{6}+\frac{9987}{86}a^{5}-\frac{9627}{86}a^{4}-\frac{3854}{43}a^{3}+\frac{9669}{43}a^{2}-\frac{10953}{86}a+\frac{1903}{86}$, $\frac{318}{43}a^{14}-\frac{1637}{86}a^{13}+\frac{589}{86}a^{12}-\frac{203}{43}a^{11}+\frac{2989}{86}a^{10}+\frac{629}{43}a^{9}-\frac{13403}{86}a^{8}+\frac{15377}{86}a^{7}-\frac{3825}{43}a^{6}+\frac{10787}{86}a^{5}-\frac{9429}{86}a^{4}-\frac{4801}{43}a^{3}+\frac{20183}{86}a^{2}-\frac{10027}{86}a+\frac{778}{43}$ | 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: | \( 803.578266232599 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 803.578266232599 \cdot 1}{2\cdot\sqrt{388863829589238784}}\cr\approx \mathstrut & 0.201905770915894 \end{aligned}\]
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
3.1.484.1, \(\Q(\zeta_{11})^+\) |
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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | $15$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | $15$ | $15$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{5}$ | $15$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | $15$ | $15$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $15$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
2.10.10.7 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
\(11\) | 11.15.14.1 | $x^{15} + 11$ | $15$ | $1$ | $14$ | $S_3 \times C_5$ | $[\ ]_{15}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (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) | $0$ | $1$ |
1.44.10t1.a.a | $1$ | $ 2^{2} \cdot 11 $ | 10.0.219503494144.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
1.44.10t1.a.b | $1$ | $ 2^{2} \cdot 11 $ | 10.0.219503494144.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
1.44.10t1.a.c | $1$ | $ 2^{2} \cdot 11 $ | 10.0.219503494144.1 | $C_{10}$ (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) | $0$ | $1$ |
1.44.10t1.a.d | $1$ | $ 2^{2} \cdot 11 $ | 10.0.219503494144.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
* | 2.484.3t2.b.a | $2$ | $ 2^{2} \cdot 11^{2}$ | 3.1.484.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.484.15t4.b.a | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.388863829589238784.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
* | 2.484.15t4.b.b | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.388863829589238784.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
* | 2.484.15t4.b.c | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.388863829589238784.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
* | 2.484.15t4.b.d | $2$ | $ 2^{2} \cdot 11^{2}$ | 15.5.388863829589238784.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |